If you can wrangle a part-time F/I role and it suits your long-term ground job, it suits the flying school and it suits your home life, then why not. But you don't write that the flying schools are chasing you, you are in fact chasing them.
How close you are to your two areas of work, how old you are and what you have to offer are to be considered also. Why? Part-time positions imply short notice engagement, if you're an hour or two away from the a/c, it's not convenient for anyone. Every flight school has a truckload of pubescent-like grade 3s with the ink still wet. What they don't have is mature age greybeards, even newly qualifieds, who bring a whole new range of skillsets to current and potential students.
As a greybeard myself, I once went to a flying school to hire an Arrow for a day or two and the CFI asked if I was interested in increasing the average age of his flying school's instructors on a part-time basis; the CFI apparently recognising an imbalance. I would have loved to have done so but was heavily involved in other matters.
See what I'm saying here? Convenience, life/work balance, added value.