I used to have one of those rifles, a long action 308. Ultimately, I traded it off and picked up a short action 243 (which I promptly changed to a 308 barrel via a swap), because I had long term ideas about adding a DBM bottom metal. I never got around to the DBM, but my 308 short action Savage has shot pretty well for me on the few times I took it out for plinking.
I hate to say it, but for what a used, basic hunting rifle like this would sell for, I would recommend you use it as is, or sell/trade it to get the action you want. Surprisingly to me, at the last few local gun shows I've gone to, I have not seen many used Savage rifles available (for the $250 to $400 I'd pay for one), but I have seen a decent number of older Remington 700 short actions. These would make a fine candidate for an upgraded rifle.
The main thing I like about the Savages is that you can easily change the barrel. I do not care for the overall feel of the action, or the various changes to the short action over the years (like the bolt release and action screw spacing) that has made aftermarket parts less widespread. Some of my Savages have weak ejection and an atrocious bolt lift also, so as much as I appreciate the modular nature of the Savage rifle, I now prefer a Remington 700. When I wear out the barrel on the 223 Rem 700 I have, I'll get a barrel nut and new barrel for it. But note: I buy rifles with the idea that I'll put a thousand rounds a year through them, and in reality, it's closer to 500 rounds a year spread over 2 or 3 bolt actions.