vamp (v.)
"extemporize on a piano," 1789, from vamp (n.1) "upper part of a shoe or boot," via verbal sense of "provide a stocking (later a shoe) with a new vamp" (1590s), then "patch up, repair" (compare revamp). Related: Vamped; vamping.
vamp (n.2)
"seductive woman who exploits men," 1911, short for vampire. First attested use is earlier than the release of the Fox film "A Fool There Was" (January 1915), with sultry Theda Bara in the role of The Vampire. The movie was based on a play of that name that had been on Broadway in 1909 (title and concept from a Kipling poem, "The Vampire," inspired by a Burne-Jones painting). The stage lead seems to have been played by Kathryn Kaelred and Bernice Golden Henderson. At any rate, Bara (born Theodosia Goodman) remains the classic vamp and the word's wide currency is attributable to her performance.
A fool there was and he made his prayer
(Even as you and I!)
To a rag and a bone and a hank of hair
(We called her the woman who did not care)
But the fool, he called her his lady fair
(Even as you and I.)

[Kipling, "The Vampire"]
vamp (n.1)
"upper of a shoe or boot," 1650s, earlier "part of a stocking that covers the foot and ankle" (c.1200), from Anglo-French *vaumpé, from Old French avantpié "vamp of a shoe," from avant "in front" (see avant) + pié "foot," from Latin pes (see foot (n.)).