Just try with your TTIG - you’ve got it plugged in - it’s as simple as you going downstairs and trying it out.
And as @LoRaTracker says, no one here can make a suggestion based on the information you have given and even then, probably not at all because, as I said, we’d do a site survey.
Any which make use of the core Semtech GW chipsets! 
(SX1301+2xSX125x, Sx1308+2xSX125x, SX1302+…, SX1303+…) ie they are all much of a muchness with only minor variances based on build standard, components, temp range and environmental targets! and how well they follow or increment from the original Semtech reference designs and layouts…I repeat
Main issue is if relative position (GW directly above door sensor) is correct then the polar plot of the ant (any ant - even external ones) will show relative nulls directly above or below, reducing sensitivity - hence suggestion to offset relative position if possible. Given your new/added info above showing another building in close proximity as Stuart suggests reflections may be your freind! The only way to be sure is JFDI! ('scuse the French) - deploy and test, anything else per Nick is speculation and educated guessing…I had 1 UK client years ago who deployed to an office block in Florida US - GW and main sensors were in basement (Plant room!), with last minute need to add another spare sensor to plant room attop the escalator shaft at top of building…some 20-25 stories above! They got signal through - but suspicion was it wasnt by direct through floor connection or even signal propogation down stairwells or elevator shaft but rather by reflection from other tall office blocks a couple of kilometers away! Such conditions difficult to predict on a forum posting…
BTW where are you placing the sensor? If inside a (metal?) mailbox likely signal very attenuated because…
Now, as it happens, I was testing this just yesterday.
With a LoRa device in a small sweet tin and transmitting at SF12, BW 125Khz, 14dBm, LoRa reception in my Shed was marginal when the transmitter was in the tin on a table in the house about 10M away.
Do you still have the results from when you tested a device inside your fridge to compare with?
Its not something I have measured as such, but I recall I was doing (receiving signals from a LoRa device in my fridge\freezer) to measure the frequency drift of the SX126X devices that can operate at bandwidths down to the 7800hz minimum as they use TCXOs.
But I guess it would be useful to know how many dBm a fridge will absorb, watch this space.
What like this one
SF7 - typically -72–78dbm aro SNR8.5-11, with GWs 2 rooms away - 7-12m with 2 walls/internal windows/doors 
All three fridges are monitored here. See similar values across them all (BTW all are ‘built’ in types so behind cupboard doors also…) Worst gives typically -82 to -88dbm - they are in different rooms/locations.
Er, no.
Unless I missed something, that just says what the reception was, not what the difference between in fridge out of fridge was ?
It was partly a joke back to Nick after his question, however, since you ask - A couple of data points below out of the fridge (and in open room) similar distance from closest GW (a TTIG) - tends to be the main receiver reporting relevant RSSI, one floor below a Mikrotik (approx 45deg up angle from node similar horizontal distance to the TTIG, 2 floors below test LPS8, iMST Lite & TTN KSGWs! The others capture/report RSSI if TTIG busy or misses and RSSI range is typicall -73 to -91 across the test GW suite
Turns out the answer is that the fridge attenuates signals by around 30dBm, which would cut the distance\range by around a factor of 30.
Thus if you were getting 5km with a sensor outside a fridge you might only get 160m with a sensor in the fridge.
Fridges might vary …
And that is probably in line of sight, not with 8 building floors in between.
Do we have a new acronym here?: YFMV 
“2nd floor” contradicts the “3rd floor” in the title.
Mailbox on the second floor (instead of basement or first floor). Does that mean the mailbox is located more deeper inside the building and not on the outside?
Below the buildings is a shopping mall which has ~2 floors. So don’t care about it, cause both buildings have same ground level. Mailbox is not deeper inside the building.
Have you tried it yet?
@mensa I did not see your response to this but the sensor’s antenna and its placement are (at minimum) of equal importance as gateway antenna and placement.
Yes, the mailbox is metal. It’s the Dragino LDS02 sensor. I think it’s antenna is not the best. Do you have any experience?
Tried what exactly?
I am still waiting for suggestions for a cheap and simple gateway which could do this. I currently only have the TTIG.
I know one posted the chipsets here, but isn’t there any out-of-the-box running gateway for beginners with much better coverage than the TTIG?
I don’t have a Dragino LDS02 myself and could not find any visual information about its internals.
It’s not as simple as that ‘throwing some better gateway at it’ will solve your problem. Several factors are involved (reread above responses and other information on the forum about antenna’s, gateway coverage and the like to get an impression).
There will be other gateways than TTIG that will provide better coverage but that will mainly be because of better, usually external, antenna. Whether a (gateway with) better gateway antenna will actually solve your problem is unclear and not guaranteed, because as said many factors are involved (of which an essential part on the end device side).
Welcome to the world of radio communication.
To start you can do some simple experiments like hanginging your TTIG outside the window/building and positioning the LDS02 outside the building (in near line of sight) to see if they will communicate in (near) line of sight conditions. From there you can start placing end device and gateway in other locations/positions (only one at a time!). It may help to get some end device with external antenna (with an antenna connector for experimenting where you can try different antennas) e.g. something like TTGO LoRa 32 v2.x or TTGO T-Beam. Maybe you can borrow a gateway from someone to do some experimenting with external antenna and positioning. If there is a Things Network community in your area try to visit it and learn from others.
For your application/situation there will probably not exist a ready solution that will work right out-of-the-box (independant of any environmental conditions).
You have a gateway and a device - why do you need another gateway to try it out? Literally ten minutes effort. Just do it. There is NO reason not to.
You have no idea what coverage you can get with your TTIG as you haven’t tried it. There are adaptions documented on adding a better / external antenna to the TTIG.
But the short answer is NO, there are no other out-of-the-box gateways for beginners - the TTIG is the closest you get to plug & play - all the rest require you to read the documentation & configure them.
The point with the list of chipsets is that all gateways have silicon from the same manufacturer - some of the chip sets are slightly less power hungry &/or more sensitive - but for most purposes are pretty much the same. The differences are in packaging (the box) and interface (web/console).
I did of course already try!! But it just does not work. When I go down to the ground floor I can get only connection in line of sight. That’s the reason why I asked here.
I also was walking around with a battery pack with the TTIG, but I was not able to get in connected with that device in that place, so that’s why I am looking for something stronger.
And the TTIG was connected to TTN how? …were you monitoring that the TTIG itself was online during that time? Connected via Wifi tethering to mobile phone? (Often works for me in right locations for such surve work)
Reality is a small sensor with an inefficient built in ant, mounted inside a metal box, talking up 7-8 floors and even then possibly not normal to the floors but also likely subtending an acute angle through building infrastructure such that the radio path through any floor or walls then appears much thicker than nominal is always going to be a bit of a hope and a prayer unless you get assists by reflections or channels through building features such as stairwells or elevator shafts… you need to try various locations for the GW (as location for the mail box is fixed!) and look to get ant signal for the sensor outside the mail box.
“Stronger” isnt a thing in the LoRaWAN world as devices are constrained to operate within Radio regulatory limits wrt emissions and usually operate at or close to those limits unless using inefficient antennas or device and GW very close to each other with ADR enabled in which case they automagically adapt to the high strength signal and reduce signal strength and minimise on air time to save battery and spectrum utilisation - note they dont adapt the other way and go higher than permitted…
Adding a higher gain ant to GW doesnt help as you then have to reduce the GW TX pwr - a zero sum game - also beam strength gets focussed more towards the horizon…not vertically to a node below or a GW above…you might consider angling the node/GW to help directionality but impact is usually limited…