His Baseball Reference major league page ends things there, though his minor league page has his career continuing in Miami for a couple of years before going to the Dodgers system. How he got to Miami is covered with a short line in an article in the May 15, 1958 Gazette and Daily from York, PA:
"Senators Sold infielder Jerry Snyder to Miami"
Right to the point! This was part of a larger group of cutdowns mentioned in the article. Miami was a Phillies affiliate at that point, and transferred to Baltimore in 1959.
Snyder goes to the Dodgers chain in 1960, but I have not been able to find how that occured.