1
general::
Instantly, immediately, as in I can't tell you how many right off the bat, but I can find out. This term alludes to a baseball being hit by a bat. [First half of 1900s]
American Heritage Idioms
2
general::
Phrase(s): right off the bat [and] straight off the bat
Fig. immediately; first thing. • When he was learning to ride a bicycle, he fell on his head right off the bat. • The new manager demanded new office furniture right off the bat.
McGrawhill's American Idioms And Phrasal Verbs