During my years on the job market the two places I accepted offers at had exploding offers. Both wanted decisions within a week and one was particularly adamant about finalizing my decision. I don't regret making a go of either offer; both have proven excellent for my career.
Coming from someone who's been tenure-track for a number of years, my gut sense is that hiring schools will be more willing to wait during fall hiring. The later you go in the spring, the more pressure a department has to finalize a hire and line someone up for their fall semester. At most institutions, fall course schedules are already posted and students need to register for classes (or will register very soon). The potential that a search fails looms over the search committee's head. The smaller the department, the more devastating a failed search will be. I've been on a search committee for a smaller department during a spring term, and this fear is quite real.
Here are the factors I'd weigh…
1) Could you be very happy at the place providing the offer?
2) Is the place with a current offer one of the top two places you're a finalist at?
3) Is this your first tenure-track job? If so, you can always return to the market after 3 or 4 years & be a stronger applicant - move up.
4) Is this a SLAC or institution with a smaller department?
If your answer is yes to all four, I'd be hesitant at passing up the opportunity. My fourth question reflects reality that at many SLACs especially, faculty fear candidates using them as "backups". So late in the hiring cycle, I can envision a SLAC figuring someone with 3 other interviews might be using the place as a backup. Indeed, they might then start talking to other finalists to avoid a failed search.
The other thing to consider with your wanting a second offer for negotiating is that at many institutions salary and teaching loads are fixed. The exception tends to be R1s and perhaps R2s/elite SLACs. But, from my experiences and that of colleagues, faculty are on a similar pay scale with everyone doing their fair share of teaching unless they come in with grant money or have an all-star research record. So, there might not be a tremendous amount of negotiation possible.
I hope all of this helps!