I had a similar problem here, with one wireless access unit unable to reach all the places I need wireless internet.
The original wireless modem router is in my study, where the desktop PC is hard-wired. That covered about half the house.
I bought a (reasonably reputable - TP-Link) wireless-N access point, with the maximum permitted power output, and put it at the other end of the house, connected by Cat5 cable.
I put it on Channel 13 since there's nothing else nearby higher than 11.
I now have good coverage throughout the house, garden, and my shed.
I had no problems whatever with several different wireless devices connecting to the two. As far as I can tell, my iPhone switches between them seamlessly. I don't walk around using WiFi, but when I've checked it may be on either of the two but working as it should.