Deer hunting is an awfully broad description of the hunting you intend to do and says nothing about the terrain. Size of deer (100lb or 250 lb) and type of hunting (stand in the woods or hiking above timberline) also would help in getting recommendations that would fit your situation.
Average size of deer runs between 165-230lbs. Terrain is both thick and open. This will be a stand rifle but need to be suited for the stalk into the stand as well. A walk up to or over a mile with a treestand and gear on your back weighing close to 30lbs is normal. Most hikes are on average 45min -1hr. Vegetation and terrain is very diverse. One minute you are on flat ground next can be barely making it up the side of a steep ridge. Visible distances can go from not seeing more than 10yds to wide open up to 300yds. If I know the stand setup is relatively thick and the shots are known to be close I carry my brush rifle which is a Rem. 760 in 30-06.
Just my opinion so take it for what it's worth. Given your criteria, I would favor staying with 308 and above for that size deer although you could certainly go smaller and I'm sure there will be a hue and cry about shooting deer with tiny bullets but that is just not me. Since you already have an -06 that would be my first choice.
Next would be rifles and again just my $.02 but I favor medium to lightweight bolt action rifles especially when you are packing a lot of stuff so about 8 lbs all up is my limit including sling, scope, mounts, and full mag. The M70 FWT is about as heavy as I would consider and I have one of the newer ones and the fit and finish are very nice. Also have a Kimber and they may need a bit of tweaking but don't be afraid of them.
Don't have any experience with Tikka but they are well regarded. Wouldn't be afraid to look at a used M700 or other brands that you could pick up inexpensively and customize and still stay within your budget.
