The "modern" National League career record for outfield assists is three-hundred thirty-nine which is held by Max Carey of Pittsburgh & Brooklyn.
Amazing trivia alert: only eight players in Major League history have ever had four outfield assists in one game and Bill Crowley of Buffalo did it twice during his career and both instances took place during the same season!
Dummy Hoy, Jim Jones and Jack McCarthy each had three outfielder-to-catcher assists in a single game for a Major League record that will probably never be tied.