Tight budget?
For a deer rifle on a tight budget you could do far worse than a Yugo M-48 Mauser. XLNT condition ones will cost you well under $200. They are 8mm Mauser, which is comparable to the .30'-06 performance-wise. Surplus ammo is available for practice, and deer-hunting softpoint bullet ammo is available at any decent sptg gds sto.
Use it "as issued" at first if you have good eyes. If yr eyes are elderly, you might want a "ghost ring" style rear sight which is easy to substitute for the military open rear sight.
Upgrades including better trigger, laminated Boyd's stock, scout 'scope mount and 'scope, are to be had for not too much $$, and you can add these piecemeal as yr finances allow. With the ones I've mentioned, you can do all the work yrself.
There are other milsurp bolt actions for about the same money, or less, but they all suffer from either hard-to-get, expensive ammo, or considerably less accuracy. That's why I'm favoring the M-48 for you.
There are also, of course, many other fine rifles for more $$, but you mentioned a limited budget.
AVOID the Mitchell's Mausers which you see advertised in all the gun mags. This is usually the Yugo M-48, and the 2-3 I've seen look really nice, but from other sources they are a LOT cheaper, with the same fine Mauser workmanship and design.