FlyFish is correct that, from what you have told us, your gun might be in the group of 300 M1926's that were ordered by the HPD. Unfortunately, since it has been refinished the HPD marking on the back strap may have been polished off. The only way to find the shipping date and the original shipping destination is to request a letter from Roy Jinks, the Factory Historian. The letter will cost you $50 but is important in documenting a historical firearm.
Even if the gun was not in the original 300 that was shipped to HPD, it very well could/would have come from Wolf & Klar. W&K was a jewelry store/pawn shop in FT.Worth in the mid-20th Cent. and they were one of the most prominent suppliers of handguns to southwestern lawmen, cattlemen, and oilmen. The owner may have ordered it as a private purchase.
The guns are frequently found with a primitive form of engraving and carved pearl grips.
Here is one (sn341xx) that belonged to a man who was the Chief of Police in two different TX towns in the 1930's.
And another (sn345xx)that is unengraved and with the original grips. Yours should look much like this one
Look in the ejector rod cutout to see if there is a very small "B" preceeding the serial number. That would indicate that the gun was blue when it shipped from the factory. Another one in my collection (sn343xx) the same 100 as yours is blue.
I don't think that it has been said but the same serial number that is on the butt should be in the ejector rod cutout, on the back of the cylinder, on the under side of the ejector star (difficult to see...look through the chambers) and on the inside of the yoke (only seen when the yoke and cylinder are removed).
Do all that you can to find out the history of the gun and preserve it. That means getting any documentation from the previous owner or his family, including photos, newspaper clippings, badges, etc.