Pull ups are one of those exercises that can really be used as a barometer for training success.
If I were you, I would try and mix it up. When we are talking the type of goal that you are after we aren’t just talking about increasing your power production but also ability to have strength endurance as well. So that being said let’s use the SAID principle and periodization to the best of our abilities.
SAID calls for adaptation to specifically apllied imposed demands, which essentially says that your body will change if given the correct reason to change. Your goal is really to do more pullups (to have more pull up endurance) Considering you have the ability to do 7 already, a big piece of what we are talking able is you having the energy to do more. With that being said, lets start with position and technique.
There’s a little more to pull ups then just grabbing the bar. You wanna create a maximal amount torque. To do this, I would start or by grabbing the bar one hand at a time in a fully flexed shoulder position. To create the necessary torque, twist your thumbs outward as if you are screwing your hands in. This will help you use your lats more effectively. You don’t wanna have a leak in energy. This means you don’t want your movement pattern to be compensated by using the inappropriate muscles. Creating enough toque will create a primer to successfully pull you up.
With that being said there are a few things you can do to program your pullups. Being that your goal is to increase your reps to 12, I feel as if it would be best to train in a variety rep ranges, sets and loads. Maybe try setting up three different days. Do a day of 5 x 5 reps, 4x 8 reps and 2x20reps. On the 5 rep day, you will train more for maximal strength. Try going weighted with a belt or hold between your feet and rest 90 seconds after a set. The 8 rep day is meant to be a progressive day. What I mean, is that for each workout that you can finish 4 sets of 8 reps with 45 60 seconds off, you increase the reps by one the following week. The day is for assisted pull ups. Complete all the reps and sets using either a band or an assissted pull up machine.
Hope this helps,