Build your own hotspot booster/extender?

I recently stayed in a place with no internet, but luckily in the neighborhood there was a FON hotspot. Unfortunately the signal was quite weak; I could only pick it up with my laptop; any phones did not see the network. Moreover, it was on 2.4 GHz only, and there were a number of other WiFis on 2.4 GHz, some of them using the same channel, so the connection was very unstable (and slow, which wasn't so much of an issue).
Now I am wonder whether it would be feasible to build a hotspot booster/extender for such scenarios.
My first attempt is a TP-Link TL-W1043ND v2.1 running OpenWRT. I set up the WiFi as a client to connect to the wifi hotspot (as wwan interface). The idea is that it could be placed in a suitable location which gives the best reception. I then connect to it via LAN ethernet a pure access point device, preferably on 5 GHz only (to cause as little interference as possible to the hotspot's signal).
I tested that setup using a Cisco Linksys E2000 set to 5 GHz and with its DHCP server disabled, but I guess more or less any access point should do, and the 5 GHz wifi's performance is of less concern here (it needs to reach to the neighboring room only).

I am wondering whether you have some recommendations for such a scenario? Which devices should give the best reception to the hotspot? (I've quite some collection of old router/wifi hardware)
Any particular hints regarding where to place it, or is this merely a question of trying it out?
Maybe some idea to replace the antennas?
Direct line of sight will unfortunately not be possible.

I'm also wondering whether there are reasonable repeater devices for such scenario, i.e. a device which can work as a client on 2.4 GHz and an access point at 5 GHz (and preferably doing NAT in between)? This might help a bit to reduce power consumption.

Bonus question: is it possible to configure the OpenWRT client to automatically log into the FON hotspot? Of course this can be done manually from any device behind the NAT, but automatic authentication to the hotspot would be the icing on the cake.

Thanks a lot for any hints.

A CPE is ideal for mooching the neighbor's wifi since it has a directional antenna built in. But there are few CPEs for 2 GHz, and none are dual band with gain antennas on both bands (you'd have to carry two separate devices).

But in general if you can see a signal at all on a laptop, it will be quite usable through a desktop router like a WR1043.

The configuration you want is called a routed client. It is not possible to bridge a regular AP-STA link originating from an AP that you don't control. SO the routed client connects the WAN network to the AP and uses NAT to hold one IP address on that network and appear as a single user to them.

The 'travelmate' package is very useful for selecting different AP networks and connecting to ones that are available and ignoring others as you travel.

Thanks for your feedback. The problem was, it was barely usable with the laptop.
So I am wondering which router would give me the best results, i.e. which criteria/features to look for.
In principle the configuration worked with a stock OpenWRT and was trivial to set up. Will the "travelmate" package provide me with automatic logon to the hotspot's captive portal? The description gives me some hope, but I guess I'll need to figure it out myself for FON.

Travelmate doesn't automatically log in. There are many different captive portal schemes and some are written to discourage automatic login.

For FON Hotspots operated by Deutsche Telekom customers the following script works:

I just need to figure out how to best integrate it into the OpenWRT configuration.