There are some who would say that Buddha did not teach reincarnation but instead spoke of rebirth.
Reincarnation, literally "to be made flesh again", is a doctrine or belief that some essential part of a living being survives deathh to be reborn in a new body. This essential part is often referred to as the spirit, soul or "I".
Rebirth refers to a process whereby beings go through a succession of lifetimes as one of many possible forms of sentient life, each running from conception to death.
It is important to note, however, that Buddhism rejects concepts of a permanent self or an unchanging, eternal soul, as it is called in Christianity or even Hinduism (and therefore reincarnation, which require it).
As there ultimately is no such thing as a self (anatta), rebirth in subsequent existences must rather be understood as the continuation of a dynamic, ever-changing process of "dependent arising" (Pratītyasamutpāda) determined by the laws of cause and effect (Karma) rather than that of one being, "jumping" from one existence to the next.
So, sorry, no reincarnation for you, unless you plan on being a ZOMBIE.
May the force be with you