Tall rider - good cockpit/body position for downhill
Are there other aspects that are causing you to position yourself wrongly on the bike?

I was taught that to get the best brake angle you should stand behind the seat, feet on the floor and reach for the bars. Extend your fingers forward to continue the line of your arm and the lever should be rotated to touch under your fingers.

I'm a short person so I'm not that qualified to say how a tall person should alter things but beware of over-reaching. As mentioned before, you'll get a more direct feeling from your steering with a short stem but a little less stability. I wouldn't go for a longer stem personally.

You may be able to raise your bars a little if you have spacers above your stem. Failing that, so you have a bike shop locally that would let you borrow an ex-demo or second hand bar? Could you try some bikes out in a shop to get a feel for reach? Some shops have turbo trainers that could at least give you and idea of how different frame sizes feel.