Josmund,
What's your budget? What's your caliber range?
You say you don't want to spend a ton of money, but if you could verify what type of budget you're hoping to stay within it would be easier to guide you to some products.
You say you are interested in a smaller caliber ... but if you could be a little more descriptive of your range of caliber size that you consider to be "smaller" that would help also.
Is "smaller" anything below .284? Below .264? below .257? What is small to your thinking?
If you would also consider (as Father Forkhorn suggests) a military surplus rifle then your options would have to include anything that was "small" enough to satisfy your "smaller" requirement. I'd look into the 6.5 x 55 Swedish rifles. You can still find them on the racks in the stores, though they are getting harder to locate. Father Forkhorn's suggestion of the 7 X 57 option is also a good one, if that is "small" enough for your liking.
Budget? Caliber range? Is Military Surplus a viable option for you?