Reworking the Springfield would surely result in a true "classic" .30-06....particularly if stocked in the style of a 1910-1920 era sporter (something like Roosevelt carried on his famous safari).
Another good custom build would be a .30-06 built on a '98 Mauser action with an early British style sporter stock. I am a dedicated Mauser guy so this would be my own choice and the earliest Mauser sporters (custom and commercial) appeared very shortly after the first custom Springfields came out.
The Winchester Model 70 actions came out in 1936 and are too late in the game to attract me to one as a "classic" .30-06, but many do consider the Pre-64 Model 70 as the "classic" American Commercial rifle...so this too could work for you.
The Enfield 1917 or Remington Model 30 could work for a classic custom rifle, but like the Winchester 70 are a bit too late for my own taste in a "classic" .30-06.....and lack the Model 70's over-all reputation and popularity.
If you are wanting a classic "style" .30-06, but are not concerned about actual historic actions, you can't go wrong with the Ruger 77 (MkII, MkIII, or Hawkeye) as they have all the classic features you would want. In reality the stock style plays as much if not more part in the "classic" look of a rifle than the action does anyway.
Those would be my choices, Mauser 98, Springfield, Model 70, Enfield/Remington 30, and Ruger....in that order. All except perhaps the Model 70 would require custom stocking to become a "classic", but all would work.
