Most headphone access plugs are cheap/poorly designed op amps. To get the most out of phones...you need a dedicated amp.Pretty sure he was asking about getting a dedicated amp.
I don’t have a specific recommendation as most of my headphone amps are out of that price range, but Drop has a number of headphone amps under $1K that come up. https://drop.com/top-list/Amps/type--Headphone/max-price--1000
Head-fi is a good resource for reviews as is innerfidelity. I’ve heard a lot of good things about Woo Audio’s tubed headphone amps, but not sure if they have anything in your budget.
If you’re patient, you might pick up a used Auralic Taurus in your budget range, that’s a great amp (not tubed though), but very smooth, detailed and just a little on the warm side.