As a bass player, being in a cover band can be a good education and can really add to your musicianship if you use it to your benefit. If you're learning a tune with a part that is out of your comfort zone, it's a good exercise to learn it and analyze what elements of it work, and what doesn't work, and what you can incorporate into your own approach. Also, and this may not be as relevant with country, if different techniques are used on the tracks you're covering (i.e. pick, slap, doubling with synth, etc.) check all of that out as well and learn it if it's not already part of your vocabulary. Also, listen to the whole track and figure out how everything fits together.
Re. the OP's question, it's up to you to figure out whether playing the original part verbatim is important to make the song work, or if you can take liberties or even improve on the original part. It also would depend on how strictly the other people you're playing with are sticking to the original.
Some of the most successful touring or session bass players I know came up playing in cover bands, and I know that experience played a part in adding to their toolboxes.