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So I had an internship at a no-name boutique that I knew I didn't want to be at, and lateraled to a larger, better known boutique that pays street pay. Generally, I would give this advice in terms of timeline (which was given to me by older ppl who were even more successful getting upper-tier MMs and BBs).

1. Network HARD near the end of the spring semester going into the summer before senior year. The main goal here is just to get your name out there as soon as possible, and hint that the place you are going isn't where you are gonna want to be (most likely). It's important here to be political about it, where you are hinting at the fact you want to lateral without making it a foregone conclusion (it just looks bad/immature to preemptively say without a doubt that the place you are going isn't good enough for you, better to get there and then come up with reasons why you aren't a fan).

2. After you complete those contacts in the spring and enter into your summer internship, don't network and FOCUS ON GETTING THE RETURN OFFER. This means staying late, giving zero indication you want to leave, doing good work, etc. The single most important thing is to get that return offer so that you can say in interviews later that the bank wanted you back, but you wanted a "greater challenge" or "a more intense learning environment" or whatever. It also makes sense to not reach out because the banks don't know how many of their interns are competent yet (and thus how many spots will be open), and it looks premature to already be looking to jump ship when you've done 5 days of an intern program. 

3. From the perspective of the banks, they start to get a read on how many interns will get return offers by weeks 6-8 or so into the 10 week program. At this point, it makes sense to reach back out to the people you talked to in the spring and say basically "Hey, I am enjoying doing IB but I am interested in exploring other opportunities yada yada yada". I would also recommend being interview ready before these networking calls in the late summer because sometimes those calls almost act like 1st round interviews.

4. From here on out (until around mid-October), it's go time. Pretty much all of the spots the banks need filled are filled during this period, and it's important to be scouring websites, handshake, and your connections for any jobs that open up. This is where luck comes in, in that there's a decent chance that nothing pops up. THAT'S OKAY. Your life won't end if you have to work at CIBC/SMBC/MUFG for couple years. You can lateral later, and if you are genuinely cut out for a better spot you'll be in a small pond where you can excel relative to your peers. 

Obviously following that timeline worked for me, but that's just anecdotal evidence, and there might be better ways to navigate the tough process of rerecruiting for FT roles. Hope this helps!