I used to do the same thing, I even had thumb rests made and installed on my MM SR5's.Then on day I played and bought an Ibanez ATK 305.
No place for a thumb rest without really compromising the look of the bass. So I changed to what they call the "floating thumb". With it your thumb lays on all the strings below the one you are plucking. You move your thumb across the lower strings as you play. You are always muting the lower strings. It will feel weird for a while but it is all I do these days and it makes switching basses a little easier.![]()