From what I understood from "The Shibboleth of Feanor," (HoME XII) it was the rule that Elves were reincarnated (or perhaps re-embodied is the better term), not the exception.
The Eldar are bound to the earth, and because of this, being in a spirit (fea) state is unnatural, and something to be tolerated for as short a time as possible.
As for specific examples, the Silmarillion makes it quite clear that Finrod was quickly re-embodied:
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They buried the body of Felagund upon the hill-top of his own isle, and it was clean again; and the green grave of Finrod Finarfin's son, fairest of all the princes of the Elves, remained inviolate, until the land was changed and broken, and foundered under destroying seas. But Finrod walks with Finarfin his father beneath the trees in Eldamar. -- The Silmarillion, "Of Beren and Luthien"
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The Shibboleth also makes it clear that Finwe was to be re-embodied at some point. Speaking of Finwe and Indis' daughter Findis -
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She did not go into exile, but went with her mother after the slaying of Finwe and they abode among the Vanyar in grief until such time as it seemed good to Manwe to restore Finwe to life.
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Hope that helps!