For many years, my dad had a pharmacy in a small town in Maryland. It was not unusual for him to get an occasional late-night phone call from a patient asking him to come open the store and fill a prescription on an emergency basis. He was glad to do so, but he always did two things first: a) he called the local MSP barracks to ask a trooper to drive by and check on him at the store (which they were glad to do, BTW), and b) he got his 1903 out of the nightstand drawer to take with him. According to Dad, the troopers knew he had the gun and never gave him any grief whatsoever. He carried it in his jacket pocket at first, but one of his friends gave him a belt holster for it (when I was about 7 years old, IIRC) and he used that thereafter. I have also occasionally carried it myself, in that very holster, when needed.
In any event, he had a lot of confidence in that 1903, and he shot (and still shoots) it fairly well. Per the serial #, it was made in 1915, and it's a little beat up cosmetically, but I doubt we'll ever get it refinished. Mechanically, it is in superb condition and has never failed to feed, fire, or eject. I hope I'll
be in that kind of shape if I make it to age 97!
To the OP, my suggestion is that you do get a holster for your 1903, and keep right on carrying that handy little piece.