In folk song it is the version not the song itself that benefits from a particular introduction.

Martin Carthy's version of "Famous Flower of Serving Men" would be one (as I think mine is also recognisable although not a patch on his).

My late wife's (Jacqui Walker, nee Turner) intro to "Nottamun Town" was also instantly recognisable. Over 5 years after her death I am still trying to get it right!

Quite a few of Nic Jones's would qualify - some on contemporary songs as well as folk songs.

Likewise many Renbourn, Jansch, or Graham (often contemporary not folk).

One of the very few piano parts I can stand (OK, not a guitar intro, a piano one) on June Tabor's version of Hughie Graeme is an instant snap to attention too.