In the classical guitar world, fretting the bass string with the thumb is frowned upon. This is understandable for at least two reasons: classical guitar necks generally are wide, making it hard to fret with the thumb and fingers simultaneously; and Segovia figured out how to play the great repertoire without it.
But with narrower neck guitars and long fingers, fretting the sixth string with the thumb can offer a guitarist a decided advantage. A main one: guitarists can fret five-note chords that classical guitarists can’t get. Another one: in ragtime styles in the key of C, a barre F chord is simply unacceptable – it’s too cumbersome to get to from the C and G7 chords used with it, and it doesn’t allow for easy access to the open treble strings for melody notes.
If you are willing to attempt fretting the sixth string with your thumb, try this: Play your default fingerpicking F chord by fretting the sixth, fourth, third, and second strings only. Don’t pick the fifth string (have your thumb pick the sixth and fourth), and fret the first string with the index or little finger, but only if the music requires an F or G note in the melody. Place your fretting-hand thumb directly ON the first fret wire, not behind it, as you normally place a fretting finger. This may mute the low F note slightly, but that sound is inherent in the “Travis Picking” style, and it may give your thumb a slightly better angle to successfully fret the bass string.
Another suggestion: When you play thumb-up-over-the-neck F, try jamming the curve of the back of the neck completely into the webbing of your hand between the thumb and index fingers. This likely will place your thumb in a position to contact the string with skin from below the first knuckle. With a slight pull backward into the guitar neck with your hand, the skin will pull the string down onto the fret wire. This may alleviate any serious stress on the thumb knuckle if you try to bend it at 90 degrees to depress the bass string. Don’t hurt yourself by putting too much stress on the thumb knuckle!
Hope that helps!
— Mark Hanson