WW,
We are on the chase. You asked about the verb "to gig". It clearly comes from the noun "gig" defined above.
Murray L. Pfeffer says on his web site in discussing gig's entomology says,
"'gig' --The musician's engagement probably derives immediately from the 'gig' that is a dance or party, but'gig' and 'gigi' (or 'giggy') also are old slang terms for the vulva; the first has been dated to the seventeenth century.".
Of course, you may have already guessed that someone would throw sex into the issue. If the latter statement is true, what does it have to do with a job?
Roger in Baltimore