Una, why don't you give the J. Crew vintage cotton tees a try? They work for me. They are made with a very thin cotton that is sort of textured, but somehow that softens the bright whiteness of the white. They are very slightly see-through, though. Don't know if that's a deal-breaker for you. I kind of like it. I wear mine with a charcoal bf cardigan or a black or grey blazer, which allows for a "bit of white" (which, to my eye, has a brightening effect) without an overwhelming amount of white. Like you, I don't feel like white is my very best colour, but a little bright white in an outfit often adds that "extra something". (I also wear white collared shirts under a fine-guage wool sweater or sweater vest, but it doesn't sound like you're interested in going that route).
Oh, and I size up for a fluid fit. And I can't wear the v-neck, because the v is too low on my short self and I can't stand the feel of a camisole under a tee shirt. The tee shirt has to be the bottom layer.
BTW, what's a HEWI?