attach a second router to my network to expand coverage?

"Yes, you can do so. Connect the second Linksys router (Router2) to the main router (Router1).

You do not have to change any settings on Router1, but assign Router2 a static IP address that exists on your subnet, e.g 192.168.1.2. Turn off the DHCP server on the router, and have it use either a different SSID/key or the same SSID/key.

The key thing to do here is to turn off the DHCP server on Router2, and also to have it exists on your network as an AP (Access Point, not router!).

Give it a go, it should work.

P.S I'm not too sure how far you can go with a Cat-5 cable, but usually for SU folks, we use WDS to extend our wireless network coverage. Search for keywords ""WDS, repeater"""