Look at American Express - you can transfer their Member Rewards to Avios. I just signed up for a Premier Rewards Gold card with a bonus of 50k MR after $1000 spend in 3 months.
You can also transfer points from hotel miles into Avios. Watch for bonus deals with Marriott, Hilton, IHG, and maybe others. I think I saw a 70k offer for Marriott that would get you 25k Avios. The current IHG 70k could get you 14k Avios.
With cards like these, I plan to use them only for the bonus, and cancel the next year when the annual fee is due.