Without a doubt, you can definitely get improved sound beyond those small 2 way speakers within your budget.
I always recommend trying to find efficient (high sensitivity 90 or above), so less amp power is needed. That keeps the amp size/weight/cost low and improves placement options, that may involve receiving remote signals.
I do not like ports, if so, front firing.
Also, efficient speakers make it much easier to try tubes, now or in the future.
How large is your listening space? What volumes do you like? makes a difference of course.
used or new. You can certainly get ’more’ buying used, IF you know they are in great shape. Shipping cost is an issue buying used, try to find something close enough to pick up and perhaps listen to before final purchase.
A great deal, add shipping cost is not a big deal, however who will package them? Original boxes?
One way is to use UPS’s ’Pack and Ship’ service. Seller simply drops them off unpackaged, UPS packs and ships, they are responsible for successful protection. Get size and weight from seller or specs on-line, get a preliminary estimate from UPS, decide, then upon receipt UPS gives you an exact price. I’ve used it several times, always successful. It is a way to get a ’local pickup only’ seller to simply drop them off.
hifishark.com; search for speakers, then set a price range, I would put in $5,500 max, you want to know what is available if you stretch your budget a bit.
Sort by price, or sort by first seen new first, keep checking until a great choice pops up, jump on it.