I was going to suggest the Phil Rickman books for the Merrily Watkins series, although then you said no to religion and she is a vicar.
That said, the woman never seems to be in a church and most of the religion has been paganism rather than christianity so far. They are more crime and people-are-strange type books than religious ones.
She's the Deliverance Consultant (exorcist) for Herefordshire. I've just started the fourth book and so far have enjoyed the series.
I haven't read any of his other, non-Merrily books yet, but some of them sound quite good too and might not have the religion angle.
I've got something on my wish list called Frost Hollow Hall by Emma Carroll which might fit your criteria, although obviously I haven't read it yet.
It's set in winter, says it's historical fiction, definitely has a ghost, and she's written at least two others that sound similar. They are children's books though, seem to by aimed at ages 9-12.
I've also got one called Wolf Winter by Cecelia Ekback wish listed, and I don't think this one is for children, which is about two sisters in a remote place finding a body. Their father is away, their mother won't let the matter rest and though they say the man was killed by a wolf, what wolf cuts a body so cleanly?