Muting on bass is a bit of an art, and it's hard to come up with absolute rules as to how it should be done. The most common techniques are:
- as you say, using your thumb to mute lower strings
- gently lifting your left hand finger but leaving it in contact with the string when the note has finished
- using the fleshy bits of your left hand fingers to mute higher strings (while you fret lower strings).
I think it's that last technique that is most relevant to what you want to do. You say in your comment that you don't want to use your left hand because...
it is taking me more effort than it should, to press down a string so that it doesn't buzz
But that doesn't sound right. Just fretting one string on a bass is easy, and should be possible to do with a light touch, on any string, at any position (including around the the 1-5 fret region, where the action will usually be at its lowest). I wonder if your instrument needs setting up properly? Do you have a teacher, friendly shop, or bass playing friend you could talk to?
know that some bassists use the pinkie to mute strings...
Yes, as you get more experienced, you instinctively start to use spare fingers and thumbs to mute where you can. You often need to mute with both right and left hand to get a really clean sound, so don't get stuck in the mindset of only muting with your right hand (even though it may sometimes be useful to be able to).