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To be honest if the Linksys router was powerful enough to give you the signal you want then we can probably use it as a WAP (Wireless Access Point) to do what you are wanting without you having to purchase any new equipment.
What you will need to do is connect a machine hardwired to one of the numbered ports on the Linsks, Once done can you then go to http://192.168.1.1 and then login to the router by leaving the username blank and entering admin (all lower case) as the password.
Once logged in can you look on the setup screen and then change the Local IP address of the router to 192.168.1.1 and then disable the DHCP server and then save your changes.
Once that is done can you then wait 30 seconds and then power off the router and then connect a line from either your switch or the Uverse modem to one of the numbered ports on the router (not the internet port) and tehn power up the router and let me know if your wireless machines can connect to your linksys network and get online or not