You have a ".22/32 Target Mod 35". It's considered a '3 screw' (in the side plate) version built on the Model of 1953 NEW I frame. It likely shipped ~ 1/59. By 1953 they had evolved to all have square butt grip frames and 6" barrels with a Patridge front sight w/o exception, and were built in far lesser quantity than the more common .22/32 Kit Guns.
By Oct 1960 it was changed to being built on the J frame and became the Model 35-1. The J frame version has an 1/8" longer frame (horizontally) allowing for an 1/8" longer cyl that could be chambered in 22 Mag.
When discontinued in 1973, except for about 5 years during WWII, they had been produced since 1911, or 57 years almost continuously.
Not too many years before discontinuance due to low sales, an optional factory target grip and wider target trigger were offered by the factory:
Both shown here on a .38 Chiefs Special Target:
Smooth factory Target grips:
Photo credit: Bronco45