I haven't shot the 25-06, but from what I have read and personal experience with other calibers I would go with the 25-06 since you already have the 300. I wish I would have bought one before I bought my 30-06.
I own two of both first there on just about the same footing as far as killing performance in over 25 years of switching back and forth since you own a 300 it goes to the 2506 I have used 100 grain and up to 120 grain bullets of most common manufactures of the last several years I run 115 Nosler partitions we hunt gas pipelines with the possibility of shots of excess of 500 yards with timber cut overs that vary from 10 to 15 years I hate to track one in them there a tangle we hunt bob cats and coyotes in the off seasons it’s just the ticket I have a few kills at 400 plus yards which is good for me I run 130 grain Nosler partitions in the 270 I can’t tell in the performance not twins but very close I would look at 24 inch barreled 2506 I see the 22 inch to be quite common and it not of my liking I may be looking for another 2506 my grandson likes my Winchester 70 and has been using it for the last 3 years so I like to fill the opening
I'm surprised at the amount that like the 25-06 and it seems the 100-117gr the most popular. I don't see a big difference on paper between the two with similar weight but the 25 looks fun with the lighter weight on small critters.
I agree on going with the .25-06 since you have a .300 for the big animals. The .270 is great for deer and the like, but since you mentioned predators too, the .25-06 is perhaps the better cartridge for a yote/deer combo round due to more variety in bullet choices. In my opinion, the .25-06 is very, very close to being the perfect deer round. I've shot 90 grain bullets out of mine with great accuracy, and no coyote will survive that bullet, even way out there if you put it where it needs to be. (I don't use those 90's for deer, I use 100 grain TSX) I have a .25-06 and a .300 Win Mag, and I'm really tempted to carry the 25-06 all deer season this year. Such a joy to shoot. I really believe the 25-06 is the one to fit your specific wants at this point.