Just accepted an offer. When is an appropriate time to start looking for a new job?

Hello, so I spent two months job searching after being laid off from my full time role in December. I had quite a few interviews but never got past the screening stage majority of the time.

I was primarily going for full time roles, but ended up accepting the first job offer I received, a part time role at 3 days a week. I gotta pay my bills somehow and did not want to hold out for a higher paying position in this job market. The pay is enough to cover my expenses and have a little left over but I'll have to live very frugally if I want to stay super afloat.

When is an appropriate time to start searching for a new, higher paying job? Do I just rough it out for a few months? I am considering picking something up in my two days off to earn extra money, but I don't know what will be realistic for me. I don't really want to mess around with side hustles when there's no guarantee of earning anything.

I'm kind of looking forward to the change of pace that a shorter work week could bring tbh. Plus the office is not far from home so I won't be spending anything on petrol. My old commute was an hour one way.

I was upfront in the interview that full time is ideal for me, and they mentioned the potential to transition to full-time. But nothing is guaranteed is it?

Not sure if this was the right move but was really scared about the prospect of eating away at my savings and job searching for another 2, 3, 6, 12 months - who knows what would have happened?