So if you are having issues muting the smallest string when you are playing chords, I'd say your thumb position is too high, if you lower your thumb a bit behind the neck, you will find the is more room for your fingers to curl.
Yes this is completely different to what I showed in the picture as what I was showing was how to mute the low E string when you are playing an A minor.
It's something the more you practice your fingers will get the hang of the positions they are ment to be in, once you have the chord changes comfortable, then you can start trying to mute the low E string when you do a minor.
Taking photos one handed isn't something I'm that good at, but I tried to show the front and back of the neck for a minor and E major
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Go slow, you'll get it !