I had a company-sponsored HDHP, which included a company-sponsored HSA. Both I and my employer cotributed. 

The first contribution to this HSA came two months after I started with the company; it was, in fact, made on my last paycheck. It amounted to ~$1,300.

My HDHP insurance with the company ended on 5/31/18. 

During the time I was covered by the HDHP, I had a heart attack. My expenses for the heart attack are in the ~$4,000 range. 

I am also over 55 years old, so I can contribute up to $4,450 per year.

I know I cannot contribute to the company-sponsored HSA; however, can I open a separate HSA through a bank and contribute to it then pay the remaining $3,700 I owe providers for their services to me during my company-sponsored HDHP?