Please be careful when using a hard money lender. I purchased a rental by the means you've stated. I purchased using a hard money lender and was unable to refi out of the loan. Since I was unable to refi, I paid the hard money lender cash for the property.
A few years after paying off the loan I was applying for a refi on a different property and that's when I learned the hard money lender sold my home to another company after I paid it off. I found out cause of the credit check buy the refi company. They seen I was paying taxes on the property and asked questions because they didn't see a mortgage on my credit report.
I told the refi company I paid off the loan so a mortgage wouldn't show cause i owned the house free and clear. They then checked with the hard money lender who stated they sold the property to someone else. However, the hard money company has my payoff on record. If the hard money company thought I still owed them on the mortgage I would have received mortgage statements and delinquent payment notices from the lender. I've never received anything after I paid off the mortgage. Now I'm trying to fight to get my money back.