Lotsa old .303s on TradeMe which were a favourite of generations past... or past due
But honestly, like other have said, glass is a really important part of the equation regardless of where/what you're hunting but the choice has to factor in both of those things. Do some poking around on here, ask seasoned hunters to see what they recommend and get out as much as possible with friends to figure out what works well for your chosen game in the area you're most likely to hunt. Once you have an optics setup that you think you'll be pleased with you can reach into the piggy bank and see what's left for the rifle as the biggest factor in a person's ability to connect downrange is often dependant on experience rather than hardware - within reason of course.
I don't have any personal experience with the Savage Axis but I have a few of their other rifles, including the Stevens 200, which was part of their budget line before the Axis came out. The Stevens is my go-to gun for deer and with a bit of extra work it has shot just as well as a factory Tikka. However, the fit/finish of the Stevens has never been as good, the increased price point on Stevens rifles in NZ and potential difficulty of finding aftermarket parts could make all of this a bit of a moot point... Having said that, just try and get your hands on whatever you can and pick the one that suites you best.
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