I'd absolutely start with Winchester Deer Season XP or the Copper Impact version if you prefer copper. Both are 130gr bullets that are very accurate in lots of different rifles.
They will hold up to close in shots just fine and with the velocity the 270 Win provides, both are just hold on fur out to your 300-yard max range.
I have seen so many good results from these bullets (including some of my own!) that it is hard for me to recommend anything else for an inexpensive, accurate, effective deer bullet.
That being said, any decent bullet 100-150gr will work just fine on whitetails. Find what your rifle likes best and go have fun.