If the donor action were a standard length and weight (eg, Win M-70 or Rem M-700), I'd consider either the 35 Whelen or the 9.3x62 depending on whether you want a bit lighter bullet (225 gr, .358") or heavier bullets (eg 286 gr, .366"), as the terminal ballistics are indistinguishable from each other. Modern bullet construction makes the exact choice largely moot, as once you find a bullet weight and type, you're likely to settle on it - mediums tend to be specialist rifles for a specific range of hunting situations, like Africa - at least IME. Additionally, if you choose to rebore, the chances are high that if it's doable for one, either bore would work, provided your current barrel is not a very light contour.
However, you're starting with a pretty hefty action in the Enfield, so it makes little sense to me to go with a 338-06 or even a 35 Whelen. If you're leaning to a metric, I'd go with the 9x64 Brenneke, even though it'll need opening the bolt face, it's a step up from the 9.3x62 - heavier bullets can be pushed faster - and knocking on the door of the 375 H&H.
I consider typical mediums to be true mediums starting with the .375 and better yet, the .416s and 404s (.423"). Depending what use you have in mind, with your 1917 action and your stated intent to re-barrel, I'd choose a medium medium bore (.375") or a heavy medium (.416 or .423"). OTW, a good heavy bullet in the -06, or a rechamber to a 300 H&H or 300 WM, would seem to make sense.
Guess it depends on the level of lunacy you're ready to entertain.
If you've truly been afflicted by medium bore fever, you're entering a world of incurable possibilities -
medium bore mania.
