Scales to Play Over C6
Every chord is a subset of several scales: any scale that contains all of a chord's tones fits over it. Pick a chord to see the matching scales from its own root, the major keys the chord belongs to, and full fretboard diagrams for every scale on guitar.
A C6 chord contains the notes C – E – G – A. Any scale that includes every one of those 4 tones fits over it — from the root C, that means C Major, C Major Pentatonic, C Rock 'n' Roll, C Lydian, C Mixolydian, C Dorian Bebop, C Mixolydian Bebop, C Half Whole Diminished and C Gypsy Major. Start with the first scale listed: it is the most common choice, and the others color the same chord tones differently.
C6 is also diatonic to 3 major keys: C major (relative minor Am), F major (relative minor Dm), G major (relative minor Em). In a song from one of those keys you can also solo with the key's own major scale or its relative minor pentatonic across the whole progression, not just this chord.