Well actually, there is more than one "normal way", which is why I asked. Your way actually seems to be kind of a hybrid of the ones I use and see most frequently. They are:
Middle on 6th string
index on 5th string
ring on 1st string
OR...
Ring on 6th string
Middle on 5th string
Pinky on 1st string
Not that there is anything wrong with the way you said you do yours, but it makes a difference in terms of plotting your course from one chord to the next.
Now, in general, if you are headed towards a chord whose bass note is a fretted note (G, C, F, for example), move whichever finger is going to be on that bass note in that next chord, first.
Next, look for any "connections" between where you fingers are placed in chord #1, vs. where they will end up in chord #2. Does any finger remain in the same place? If so, make sure it doesn't lift as you switch chords.
Does any finger make a very short move? (Like, one string over, same fret you are in w/ chord#1? Or, 1 fret higher, same string?) If so, make that your next move so it can serve as an achor point for the rest of the chord coming together.
Try these things and let us know how it goes.
Good luck! - Lisa
Lisa McCormick, GT Instructor
Acoustic, Folk, Pop, Blues
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Acoustic, Folk, Pop, Blues
Full Catalog of Lisa's Guitar Tricks Tutorials
Find Lisa on Facebook!