Grab yourself a copy of James Taylor's '77 album "JT" and listen to what Lee Sklar does. He never overplays but always compliments the music. All you gotta do is play like Lee. Easy!
I mostly play original music and work with lots of singer songwriters. You just have to leave your ego at the door when - at least until you figure out what's best for the song - and don't use it as a vehicle to practice your chops. (Unless an appropriate moment presents itself.)
One of my regular paying gigs is in the back up band for a vocal school that does regular performance workshops for the students. Many of the students are aspiring singer/songwriters and bring in their song to be performed with a band for the first time. I must say, if the drummer is good and also knows how to listen to the song the pieces fall into place quite easily. We usually do 25 songs per performance and only get to rehearse each one as a band 2-4 times at best. But that keeps everyone honest and working for the best of the song.
Keep that in mind.