Smart move. It depends on the fine print of your current loan. Does it have prepayment penalties or any other ugly clauses? I suggest you simply call your current lender and tell them you are thinking of doing and see what they say. They may even give you a break on document fees and closing costs to stay with them, although it may not matter with FHA.
I just built a new home, closed in Sept for 6.34% and refinanced with a different lender for 4.75% in April. Tip - you can fold the closing costs and doic fees into the new loan if you don’t want to lay out the cash. Your monthly payment will still be much lower.