‘40s-‘60s YA/Adult novel romance suspense adventure American woman on holiday in British Isles

Looking for a young adult/adult library book I read in late 70s - early 80s. Privileged bitter American young woman (16-22) spending summer holiday (‘30s-‘60s) from her elite (girls?) boarding school alone somewhere in British Isles (Scotland, Wales, Cornwall, or maybe Ireland), published 1940s - mid 70s. Main Character named Gillian. It’s a suspense/adventure/romance which I recall having the same feel as Mary Stewart’s suspense/adventure/romance novels featuring strong female characters. Book cover featured a young woman viewed from behind sitting on the ground on top of a high cliff overlooking the see. I believe the background was light green fading upwards into grayish blue and woman’s hair was auburn. I believe she starts off as bitter/sullen b/c she feels like she has been sent away for her summer holidays because her wealthy parents (step-father and mother or vice versa) don’t have time for anyone but themselves. Appears to be snobbish, without any true friends from school or otherwise. She may have been supposed to join a fellow wealthy young female cousin to spend their holidays together but I don’t recall whether the cousin stays on the periphery of the story or plays a more significant role. I believe the story is something like snobbish American girl meets local assured boy/man. Some mystery or suspense pulls them together. By the end of the book she has changed into a kind, independent young woman with a man in her life who she looks forward to introducing to her selfishly preoccupied parents.

Any help is most appreciated.

Probably not Phyllis A. Whitney but I'll throw it out there.

Do you have an idea whether the author was British or American?