Keeping in mind what Amanda commented here, if the job he has is one with future potential then yes stay but if it's a Del taco type thing and the other job is starting higher/more room for growth then I don't see anything wrong with switching. Obviously idk your full situation and it sounds like maybe he's done this before, plus your looking for an apartment. I'd say it depends on the quality of job he has now. It would be better to switch to a higher longerterm potential job then stay in a low potential job.
Although that is important, I got an apartment with no job at the time because back at home, my auntie makes paystubs. I had just gave birth to my disabled child but we needed a place to live. I got a job after a month in the apartment. My first month’s rent was free so it worked out for me. I’ve been at my job for almost year this come November though so I definitely agree with that 💯
I had this chat with my husband when we got engaged. He had a history of quitting jobs when they weren’t interesting anymore. I told him if he kept this up he would stay at minimum wage forever and I would have to carry the bulk of our expenses. I told him I wanted his resume to be better and like mine within a few years. He buckled down and did it. He’s now in a career position because he worked his way up from porter (the guy who cleans the apartments and areas) to tech, to then HVAC school, and now in commercial HVAC after putting in his years in residential for the experience. It’s been 7 years since we had that conversation and he’s now making enough for me to stay home with our 3 kids. Your husband needs to stay consistent and patient. Not to mention the risk he’s putting on his family by not providing health insurance benefits from constantly going from one job to another in such a short time frame.