Are there any ways around special enrollment requirements to purchase health insurance?

Unfortunately not. The only way to obtain insurance in the immediate future is if you have a qualifying event.

If you do qualify for a Special Enrollment Period (SEP) but need something to cover you until Open Enrollment, you should consider a short term medical plan.

Short Term Medical plans are an affordable solution that will cover catastrophic events (i.e. hospitalizations, emergencies, etc.) for 1-11 months. However, they are not ACA compliant and would still subject you to the tax penalty for not having coverage. On the other hand, this is a small trade-off in the event you have a major medical expense that could prevent you from losing everything.