You are facing a harder path, but neither a non-target school nor a 1370 SAT makes the outcome predetermined. The bigger issue is that you may have fewer built-in recruiting opportunities, so you need stronger evidence elsewhere and a more proactive way to get noticed.
Start with the parts you can still change. A high college GPA, selective internships, leadership, measurable impact, and targeted networking can all strengthen the file. Our guide on breaking into consulting from a non-target covers how to create access when firms are not heavily recruiting on campus.
On the test-score question, do not let an older score dominate your strategy. If an application requires it, report it accurately, but put your energy into current evidence. The article on offsetting a weaker academic signal is also useful because the logic is similar: stronger recent proof matters more than trying to explain away one number.
Your goal is not to make every part of the application perfect. It is to make the overall case strong enough that a recruiter wants to test you in interviews.