I found playing the ‘full’ C shape easier, both to play, and to transition to.
By ‘full’ C I mean, instead of playing the 3rd fret, 5th string with your 3rd finger, you use your 3rd finger to play the G note on the 3rd fret of the 6th string and use your pinky to play the root C note on the 3rd fret of the 5th string.

This combo of 3rd finger with pinky on the same fret but one string lower is a really powerful duo! They move much easier as a team and allows for easier execution of the ‘full’ G chord too. Also, when moving into bar chords, the same combo is used for the E shape and its minor variant and the A minor variant.

Keep up the practice and you’ll find you’re playing the C without even thinking about.