OP,
A good headphone amp and set of headphones can be really satisfying. Perhaps not forever and only. But for your situation. Perfect to dive in and put together a great little system.
I developed mine over about 15 years. You can see mine under my UserID. Typically I listen to it every day for a hour.
First I would like to recommend Woo amps. I have three. I started with a WA6. Then a WA6SE and finally a WA5. The latter I got because I really wanted to experience a 300B amp. Honestly, this system sounded so good that I found my main system really lacking in warmth and musicality. I caused me to upgrade my whole main system (the most important part being the ARC DAC and ARC amp).
The Woo WA5 with top level 300B tubes is simply stunning…. Detailed, real and musical with incredible power to drive any headphones (and speakers if you want). Assuming you do not want to spend that much money.. if you choose efficient headphones like good quality Focals you can have and incredible system.
Personally, from my experience, beware of solid state headphone amps… with the speakers so close to your ears hardness and fatigue comes with inexpensive / SS headphone amps. I have owned many.
My favorite headphones are the Sennheiser 800s… but they required the power of the Woo WA5 to take control of them and feed them a very warm musical signal… otherwise they could be a bit trebly.