The following insights apply generally, i.e., there are of course exceptions. I base them on my own experience (Tier 4, transfer to a Top 30, then to a V30) and conversations with partners (at both mid-size and biglaw firms) who graduated from T3 and 4 schools. My four cents:
1) Don't put too much stock in career services offices' claims such as "we're located 18 miles from a major business center" or "we're located IN a major business center" (e.g., NY, Chi). Yes, it's better to attend a lower tier school located in or near a major market (or your desired market) and, yes, Martindale and firm websites will confirm that alumni from such schools are associates and partners at biglaw offices in that market.
But bear in mind that those associates and partners are the extreme exception, not the rule - and if they came from a T3, they were likley no worse than top 5% of their class. Big firms, for the sake of appearances, might "recruit" at a given T3, but will ultimately only extend offers to the top students (and career services propaganda to the contrary is just that, propaganda, and if someone as low as top 25% gets a biglaw job, it's either because of highly valuable pre-law experience/education or a personal connection). Of course, an NYC T3 grad will get an NYC biglaw job before the Chi T3 grad, just as the Chi T3 grad stands a better chance in Chi.
2) No matter what, having a degree from a higher ranked school would serve you well as you hunt for your second or third job out of school, and certainly as you hunt for one as a 3L or recent grad (assuming you don't land a 2L summer associate position). But note that most firms during 2L recruiting will base their decisions (to give a screening, let alone callback, interview) on your 1L grades and class rank from your former school. Yes, because your new, better ranked school would ultimately appear on your bio on the firm's website, they will view you more favorably than your former classmates at the lower ranked school. But only slightly, because your 1L grades are all they have to go on. This is all to say that, if a given firm only interviews those in the top 10% at your former school, if you're more than 1 or 2% outside of that cut-off, you likely won't get the interview, despite being at the better ranked school. (Note: if you do well during 2L at your new school, especially in some "hard" classes, that can change the situation.)
3) If you want to save money, Rutgers is a good choice and, as njlaw notes, its top 20% or so tend to land biglaw jobs. But again, remember point #2 above. Personally, though it likley places fewer grads in NY biglaw than Rutgers, I would be inclined (if saving money were the goal) to transfer to Buffalo. It's nearly a "mid T2," cheap, it's very cheap to live there, and in many ways a more desirable place to live, at least right around UB (plus there are some decent local ski areas).
4) If cost is not the driving factor, then, as noted by ronaldo699, shoot for Fordham, or at least Dozo or BLS. For that matter, don't rule out applying to other T1 schools (in the Fordham range).