If the signal is poor even when you are near to the Homehub a minihub won't help.
A few things to try-
Carry out ONE factory reset of the Homehub by pressing a pin into the recess button on the rear for about 20 seconds. This can often sort out problems.
Turn off Smart Wireless by selecting a fixed wireless channel. Channel 13 is usually the least congested. If you get a warning when you select it just ignore. It is a warning for USA where using channel 13 is illegal.
See link how to do that.
If that doesn't help you could try and get a new hub from BT or better idea would be to buy a decent third party router. As you are on ADSL broadband there are any number of good routers to choose from and they are not expensive. Just make sure it has an ADSL2 modem incorporated.
If you intend spending money, don't buy a new Homehub3, go for a third party router. Most if not all will out perform a Homehub3.
Any of the known brands such as Netgear, TP-Link etc ADSL modem wireless routers will be fine it all depends on your budget as regards the specs but even a basic one will be as good as if not better than the HH3.
If you think there is a chance of getting Infinity soon you may want to get a VDSLModem/router. that also has an ADSL modem. This would allow you to keep using it should you get Infinity.
Most are just plug and play but if you need to put anything in it will be the user name email@example.com and the password if the router needs one can be anything or just use BT.
If you get one with external aerials it will give you a better range than one with internal aerials.