Here are a few idiomatic phrases that imply a quick start to a continued action in slightly different ways.
Jump-start ( this is about starting a car and perhaps more about "re-starting" so perhaps not perfect)
Leap into ( perhaps implies getting into an action already in a dynamic state)
Dive in ( perhaps more about 'immersion' and sink or swim than forward momentum, although "diving start" might be hit another)
"bootstrap" is a different concept of 'self starting' 'picking oneself up by one's own bootstaps" was the old idiom which turned into a computing term which was a sort of start up protocol where the initial steps loaded additional functions which in turn could be used to get a full suite of functions for an operating system going.