In fact, I'd do it cheap to find the weak link, over here you can buy a pre amp for about 25 quid, and since your player was a hand-me-down and your amp is good but entry level, you won't need to worry about splitting heirs on prices.
If your player has a new/good condition stylus, and you have a decent amp and speakers, including a new pre amp, then you should be good to go, a final test would be to treat yourself to a brand new record that you know is good sounding and see how that sounds.
I have some old second hand records, (Chris rea's road to hell, dire straits brothers in arms, any pink Floyd's) that sound amazing, they cost me between 50p up to six quid.
Yet, I have paid just short of 20 quid for Ritchie blackmore's rainbow, live in Munich '77, from "back on black".
Brand new 180g and it sounds like garbage. It sounded great on DVD.
Check around the forum for opinions on new vinyl.
And preferably, try a record that you have heard on someone else's set up that sounds ok first, so you get a fair experiment
Thorens TD166 mkvi-Rega RB250-Audio Technica AT95, Arcam Alpha 3, Mission 730, Glade Scented Candles & Nescafe vanilla/caramel/irish cream latte