What fingerpicking songs on guitar use all 6 strings?

Pretty much all fingerpicking songs use all six strings. I'm assuming you have a steel-string guitar, because classical players don't talk about "fingerpicking" - it's the only way of playing that they have, so there's no special name for it. On a steel-string, the number of strings used really has nothing to do with how many fingers you use. Generally the thumb handles the three bass strings and the rest are played with one, two, or three fingers. Two fingers is commonest (I use two personally. Or one, if I drop a fingerpick in the middle of a song. The remaining finger just works harder to cover the bases.). If you want to exercise your ring finger, just use it where you feel it's appropriate, or substitute it for one of the others. There's no law saying each finger has to be glued to a particular string, although you might have picked up that impression if you've only done simple pattern picking.

I recommend simple Elizabeth Cotten or Mississippi John Hurt tunes for people learning fingerpicking.

Just because finger picking may have one think you have to use all your hand to finger pick is something many artists don't worry about.

The classical approach to finger style, of such contemporary music arrangements may require a full hand, but not so much for folk and traditional.

That is because once a player understands real cross-picking and can back pick with confidence and without thinking, it becomes a player's prerogative to provide what they want to express. What feels right and comes across more naturally.

Without the aid of a standard pick. (Some use one for the thumb. Mr. Guitar did many times.)

Think of the banjo players, turn guitar players by so many artists in our musical past in guitar country or folk style of playing. Merle Travis, for one of such root styles. Chet as well. And today's Dole Dykes.

Use the thumb, for the bass strings in conjunction with the string(s) you play for melody in pattern order. There are different rhythmic and reactions between the fingers once you come to understand the piece of music, or arrangement to be represented.

Or, realize if you want to keep it original and a song calls for just certain string, melody patterns.

(One of which is what I've played often is 'Puff, the Magic Dragon'. and just like Peter Yarrow of Peter, Paul and Mary. Same pattern.)

Watch some video of these kinds of players to really discover what can be done with little as two melody strings and incorporating the bass with the thumb.

You can't get away from that, especially if you want the full force of what a song can truly be made whole and represented well.

Any song.

Listen to the music.