There are several timelines you need to consider.
First is the NRMP match commitment. We've already reviewed that you must be in a program for 45 days to satisfy your match committment. Whether that means you can't look for a new program in those 45 days, or you just can't leave in those 45 days, is unclear. Regardless, I think it's really poor form to show up at a new program and start on day one looking for a new position.
Next is the practicality of how this usually works. Not many programs have openings on July 1, so even if you start looking then you're not likely to find anything. Some programs develop openings after July 1 because someone can't start, can't get a visa, etc. Or someone gets ill, or leaves to join another program. If you reapply to the match, there's a fixed timeline for that. if you're looking for open PGY-2 positions, programs tend to find out they have openings in December and January and then start looking to fill. Note that the NRMP is changing its processes, and trying to get programs to put these PGY-2 positions into the match -- we will see how that goes.
So, prime season for hunting open PGY-2 positions tends to be Jan - March. Problem is that your current program may be asking you to sign a new contract in that span. Remember that if you're not staying for next year, I need to find someone new, and prime time for doing that is Jan-Mar. So, you might very well need to decide to not sign your contract for the next year before you actually have a new spot.
Also remember that offering you a contract for the next year is at the discretion of your PD. If you drive people crazy with looking for a new spot, don't seem invested in the program, etc, they may simply decide to cut you loose. Then, if you don't have a new spot, you have nothing.
What you don't seem to be understanding is that residency is a team sport. You and your co-residents will be working as a group. This isn't all about what you want. You made a commitment to this program by matching there, and now you just want to walk away -- or worse, want that training but want to get some other training if possible. If you leave your program, there's a reasonable chance that they will NOT be able to refill your position, as filling positions at higher PGY levels becomes more difficult. if so, then your co-residents will end up pulling your weight.
I realize you have, potentially, had a charge of heart. But you also made a commitment to this program by matching there. Where I come from, that means something. You seem to think this is all about you. It isn't.