I suppose it all depends on how you would like to define US launch covers. If you were to limit yourself just to covers that were created to commemorate NASA launches perhaps one might argue that the earliest might be January 31, 1958 with the launch of Explorer 1. Below is an image of one of these covers. Or if you are more interested in weather related flights you might start with the covers prepared commemorating the Century of Progress Stratosphere Balloon flight of August 5, 1933 though of course those are not rocket ones. Or if you want to collect covers that were actually flown on rockets perhaps the earliest documented rocket mail flight was The first U.S. Rocket Mail flight was on July 1, 1931 at Struthers, Ohio though there are claims for U.S. rocket mail flights as early as the May 5, 1904 rocket mail flight from a balloon at McConnellsville, NY.
If you would like to be in touch with the big time experts on astrophilately as well as people like yourself who focus their collecting on astrophilately I might recommend joining the Space Topic Study Unit of the American Philatelic Society. Here is their website link: http://www.space-unit.com/
Instead of collecting indiscriminately, my suggestion is to not only join the Space Topic Study Unit, but even more importantly to buy one or more of the books that catalog the different covers and flights. That way you can keep track of what you have and inform yourself on what you should look for. These will also give you a sense of when you are getting a fair deal on prices or when you are getting taken for ride and overpaying. Some of them will also help you learn just what to look for so you will not be fooled by some of the forgeries that are out there, especially for the less common or more expensive material. Here are a few catalogs you might consider starting with. My suggestion might be to borrow them from the American Philatelic Society Library first and then decide if you want to buy your own copy or perhaps just photocopy some parts of them for your personal use. The APS Library's website is http://stamps.org/About-the-Library
"Space Covers: Dictionary & Identifier" by Bruce Cranford
"COSMOS Catalog" by Espace Lollini (he has a number of different space related catalogs, but the one that is best for commemorative covers is the COSMOS catalog. COSMOS is an abbreviation for Catalogue des Oblitérations Spéciales et des Marques Officielles Spatiales.
"ELLINGTON-ZWISLER ROCKET MAIL CATALOG: Set of 2 volumes." Volume I-1904-1967 International Unabridged; Volume II-1968-1972 International Unabridged, Plus Additional New Listings Discovered Since Publication of Volume 1. Edited by Perry Zwisler This is the definite catalog of covers that were actually flown in rockets up from the beginning and up and until 1972. You would not find commemorative covers for rocket launches here.
Here is a photo of an Explorer 1 commemorative launch cover:http://pic90.picturetrail.com/VOL23...03734437.jpg