I never recommend them, but what about any of the 4 Hilton HH cards? They provide hotel points, and none are from Chase, two criteria the OP is looking for.
I never recommend them because I have NO faith in Hilton's endless devaluations, but the sign up bonuses are sweet.
There is also a no annual fee version of Barclay Arrival that may suit OP's needs.
Originally Posted by
toomanybooks
I think the best single card for you right now is the Amex Blue Preferred. 6% cashback at supermarkets on the first $6000 in spending per 12 months. $75 annual fee (I think). Also gets you some online offers from Amex IIRC.
There are a bunch of websites comparing Amex Blue Cash to Blue Cash Preferred to Blue Cash Everyday. YMMV, but in general, the cutoff for BCP beating no annual fee BCE is about $3000/year in spending. It takes about $12k for Blue Cash to beat BCP.
I agree that would be a great choice, but OP prefers points to cash back.
If he wants cash back, and wants no annual fee, the Sallie Mae card is another one I'd suggest.