You also can tell something about the original configuration of the receiver by looking at the firing pin. If it appears to have been unchanged from the factory (nos signs of alteration of the block, etc) it was probably a .22 rf from the factory (#3 or #3F) but if it is altered from a centerfire or reversible for another caliber, or if the breech block has been bushed, then it was one of the many other models.
If you look under the barrel tenon on the inside (you need to take out the block to do this) you can determine whether you have a cast action (as most rimfires were) or the heavier forged action. The cast action has a hollow space under the barrel tenon while the forged action is solid there.
How is the rifle stocked? It should have a classic looking stock (almost a field-looking stock) if original, while many of the Ballards that gunsmiths built up for competition had more modern looking stocks designed for multi-position or straight prone shooting.
The Winchester 52 Barrel, as David said, is a very logical and likely gunsmith replacement, and could have occurred any time from about the mid-20s on. It is a classic piece of work regardless, since of course the Model 52 is long discontinued as well. I'd guess the work was done no later than the '50s, but that's just a SWAG.
Regardless of what you find, it's a neat piece of competitive shooting history. Congratulations, and enjoy shooting it!
Froggie
PS What about the trigger(s) on this rifle? Is it a single trigger or double set, and if single, is it mounted on a plate on the bottom of the breech block? If this last, it was a DST action that was converted to single trigger to make it legal for NRA competition. Sure would be nice to have DSTs though!
How about a picture?