A special format for Wellerisms called a Tom Swifty incorporates a punning adverb that modifies the manner in which the statement was related. Tom Swift is a series of books for kids whose sometimes adverb-heavy, suggested a name for a type of adverbial pun promulgated in the 1960s, the "Tom Swifties".
The only name I've ever seen for them is "Tom Swifties." They were so labeled in one or two joke books I had as a kid. Since then I've assumed they were named for the one that always appeared first on the list: "'I'm back already!' said Tom swiftly."