Volume
If you can do two or three sets of about 6 chin-ups, I'd focus on volume rather than any particular set or rep scheme. That's because it's hard to stick to any particular number of reps per set with chin-ups in general, and especially so when the trainee can do less than 10 in a set. (Increased but tolerable volume is frequently the goal when deloading and working back up or altering rep ranges when this problem is encountered with barbell exercises.)
Instead of doing sets of six with kipping, I'd focus on getting 25 strict chin-ups in every workout (or even every day, or every weekday). That might mean a set of six, then four, then two sets of three, then four sets of two, then one. Don't worry about the number of reps being consistent across sets.
Other tools
Negatives are an enormously productive way to increase chin-up numbers as well. Maybe make your last two or five sets all negatives, where you jump to the top and lower yourself as slowly as possible. This allows you to dramatically increase the pull-up volume you can achieve, and it does so in a more productive manner (for strength and pull-up performance) than kipping.
Holds are very useful as well. When my maximum pull-up number is low, or when my elbow is inflamed, I'll alternate workouts with an A and B system. On A workout days, my pull-up workout is as described above, with no negatives or holds, just focusing on volume and maximum number of reps per set. On B days, I pick a number of reps I know I can do reliably. (This is often quite low.) I do three to six sets of that many reps, as negatives, each performed with a hold at the beginning (at the top) and end (at the bottom) of the set. Sometimes I'll do holds between every rep. This slow, low-volume approach gives me plenty of training stimulus for the movement while avoiding the high-stakes intensity of going for a maximum number of reps in each set.
