I play lots of songs that were originally recorded on double bass and 80% of the rest were recorded on a 4 string bass. I have never once thought, "I shouldn't be using a 5 string electric bass on this!"
IMO it doesn't matter what instrument the original song was recorded on. You're the one playing the song/tune, so play it how you want.
As Al says, there are very few "life changing" moments in bass playing, so little steps is the way of things.
My move to 5's just meant I could play everything on the same bass. I played in an originals band and all the bass lines were written for 5 string. All the covers worked on a 5 string, where quite a few wouldn't have worked on a 4 string. For me moving to a 5 sting bass was the most positive thing I've done in 25 years.
You've got to play what feels comfortable but please don't do the "Jaco only needed 4" thing!!!
Just my 2p