I started to write a long answer weighing the costs and benefits of conducting a job hunt while you already have one and then I spotted the one fact that outweighs all the other facts. This is your first job.
The fact of the matter is, you still need to pay your dues. If you quit your job now, and start another job hunt, then you will not be moving up in your field, you’ll almost certainly be moving down. You need to establish yourself in a field before you start picking and choosing who you want to work for.
It is way better to go looking for a job when you can put six months of industry experience than when you can put none, or God forbid, actually put down on a resume that you quit your first job after a month or so. This consideration far outweighs whether you are looking for a job when you already have one or not.
Yes that can mean putting in extra hours. Yes that means dealing with a less-than welcoming atmosphere.
Let me be clear about this, if it were me and I were trying to break into a field and getting my foot in the door meant working in a dingy windowless basement for 12 hours a day, I’d say you’re better off sucking it up and getting it done than you would just quitting and going to look for another door to put your foot in.
That said, I started looking for a job almost immediately after I started working one. I was commuting 40 miles and only being home on the weekends. And I hated my actual work. So I started calling recruiters as soon as I had a free minute. Took a few weeks but I soon had a much better job, with a much better salary, with a much better commute.
In short, you need to serve your career before you serve yourself. Because it’s easy to hurt your career and not do anything for yourself.