Several manufactures make quality bullets that will work on elk, the best aren't cheap, but it's a small price to pay compared to the other equipment you will be using on an elk hunt. I'd go for something in 165gr. to 180gr. for the .06, topped with failsafe bullets, partition, or the swift A-frame bullets. Out of the .270, I'd go with the fail safe or the partition in 140gr. or 150gr. Be sure to sight in with the shells that you decide to hunt with. A 150gr bullet from one manufacturer won't necassarily hit the same spot as the next 150gr. bullet from a different manufacturer.
I've taken 7 elk with my .270, used the 150gr. partition on 6 of them, recovered every bullet, and took one large cow with the 140gr. failsafe, and did not recover that bullet as it went all the way through. The .270 is good out to about 200yrds., from there it just lacks in power on elk.GOOD LUCK!