I'm in a modern country cover band right now as well. I could probably help you out on some stuff you might be stuck on, if it's something we're covering anyway. I wouldn't be able to provide you anything nearly as exact as Jeepin_jer writes up, but I could help point you in the right direction maybe.
My main approach since bass tabs are hard to come by for country is to find a chord sheet for guitar and work from there. If you know major and minor scales, you should be able to work out the baseline. Another thing I do is watch others play the tunes on Youtube. Sometimes there are videos of just bass covers, sometimes I watch other cover bands, and sometimes I watch live performances. I've noticed on a few songs, specifically Point At You and Chillin It, that the bass line on the record is not always what the live performer is playing. In fact sometimes quite different.