You really don't want to play games on any wireless. You wireless is likely limited by your xbox it likely can't use many of the features in the router than make it go 150m or 300m. Still it is not the speed that matters for games as much as the quality. Interference is the key issue here and that is related to your house design and how many neighbors are using wireless.
Your first issue is exactly which box att is using they use a couple of options and some depends on if you buy some of their bundle crap and what market you live in.
In almost all cases you can just plug your second router behind their and either disable the wireless in the att box or actually use both using one channel on theirs for non gaming and your new router wireless just for games. The better way is to set their device to bridge mode which make it only a modem. You can then plug your router in theirs and it will run as the only router. You sometimes do not have the option to run in bridge mode if you are getting other services from them.
You may also be able to run your new router in AP mode and cable to the ATT router, If you can place it closer to your xbox you could then use this device just for gaming....but if you can get a cable all the way to the xbox that is your very best option
Hard to say if it will help since wireless is dependent on so many things.