I've had good luck just emailing S&W Customer Service (a link is on the S&W website). I give them the model and dash numbers along with the serial number and ask them for the production date.
If you have the original box, I've been told that the Spec-Ord number can normally give you the date the label was printed. The 1st digit of the spec ord number is the last digit of the year and the next 3 numbers are the Julian day of that year. For example, a 66-4 with a spec ord number of 4286 would mean its label was printed 10/13/1994. That is the 286th day of 1994 (the only year ending in a 4 that a model 66-4 was made).