Lift to get strong, don't lift for the sake of lifting
Make sure you don't add lifting workouts just to feel like you're staying busy. The point is to actually make progress, not to do busy work. Investing more time in strength training will take time away from the real thing giving you progress: rest. You don't get strong by working out, you get strong by recovering after working out.
At the moment, a month into the program, I bet you're feeling underworked. That will change drastically if you follow the program and keep adding weight to the bar every workout. That's when you'll understand why it has three weekly workouts and not five.
Off-day gym workouts
That said, a three-day-a-week lifting program can easily be made into a better version of itself by going to the gym on one or more rest days. Those additional gym visits should be spent on active rest and mobility.
For example, if you're doing Starting Strength on Monday, Wednesday, and Friday, you could definitely hit the gym on Tuesday and Thursday with something like this plan:
- Double-length warm-up (e.g. 10 minute run instead of 5)
- Full-body joint prep, rolling out each joint from the toes to the neck
- Yoga or other stretching for half an hour to an hour, focusing on your most immobile areas. This probably means a lot of attention to your hamstrings, hips, shoulders, and back. Impeccable posture and flexibility is the goal.
- A brisk or incline walk for a short while. If your goal is pure strength then your off days should be more restful, but if you have broad goals then a short-to-medium metabolic workout would be fine here. That could be a mile run, a set of bodyweight calisthenics, or something more challenging like AMRAP kettlebell clean-and-presses.