Once it becomes apparent one is likely to miss a flight, a phone call is probably the fastest and best way to secure onward transport. Thes can be done long before getting in a queue (and even whilst in queue).

The Club is a resource - one pass for one person at $50 might present an impediment to those with tickets on separate PNRs), as might be kiosks near the check-in counters and elsewhere (if any), and the check-in counters (but may also offer queues).

I use MIA because it's sometimes required for my travel; I'd not schedule a full USCBP and TSA experience type connection with less than three hours, unless I had Global Entry.