Most of the replies you'll find in this thread are coming from the look-up tables in the Standard Catalog of Smith & Wesson by Supica & Nahas. It's a fantastic reference and worth the money for anyone who has a deep interest in Smith & Wesson handguns.
In that index, it does say that the "c" prefix SN's began in 1948 (with C1xxxxx) and ran to 1967 (C999999) and it doesn't give further info.
There are certain features and details that some folks can pick at on the revolver itself to get a closer estimation of it's age -- I'm not nearly so good at that... and you'll need to upload pictures.
Even with extra details, most in this forum can simply give you a range of years it was produced, according to what is published. If you truly would like to know when it was
shipped, that information is indeed available -- in the form of a factory letter from the Smith & Wesson historian, Mr. Roy Jinks. The look-up and letter is a service available for $50 and if he's got the information (likely, he does!) he will tell a brief history on the model itself, he'll tell you exactly the features it had when it was shipped, the shipping date, and he'll tell you specifically to
where it was shipped.
It's pretty neat. It's not free, however.
In the case of very old, rare, or extremely fine revolvers, the factory letter is almost a must and can greatly increase (and/or simply "back up") a value. In that way, the $50 is an easy investment.
The Model 10, and pre-Model 10 (the K-frame Hand Ejector, Military & Police, Model of 1905, Fourth Change) is perhaps the largest single produced model of handgun in the history of man. It's been suggested some 6 million have been made over the last 100+ years.
They made so many... because it is
THAT good!
I have four of them myself.