This is the very reason why I find vinyl so frustrating, it takes considerable money to make a noticeable impact. I've read in many phono stage reviews that it's nearly impossible to find a fully capable phono amp for less than $2,700.00. I'm not so certain whether or not this is true but will say that in relative terms, investing $3,00.00 in a phono amp. cartridge and turntable would be considered 'budget entry level'.
I agree with those others who believe that your focus should be on a better phono amp. I believe the chain towards improvements goes; phono amp, cartridge, tone arm, turntable but this could be an over simplification. The question is whether or not you'll really notice a difference between your current phono amp and the one you'll be able purchase for a $1,000.00?
So if you go the phono amp route, then I'd say try buying one that's used with a rating of 8/10 or above. My phono amp is an ASR Mini Basis Exclusive and is a really good phono amp for its price range. I've seen older used models of the ASR sell for about $1,500.00.