Richer Sounds sell a Btech BT928 for £30. Can't think of anything else that does what you need that isn't either portable, £60-£70 or rubbish.
Edifier do a desktop headphone amp; the HA11. Not seen many comments on it, but those I have were not that positive. Can only be bought from China it seems, and with postage is near or over £50.
Not sure you are going to see any improvement in sound quality. Not for what is a low price for a desktop headphone amp. There are some mini class T amps in your price budget, which you might see if you are looking on the famous auction site. Some have a headphone socket. They are primarily designed for speakers though. The ones that do have a headphone out, it's there for convenience only and offers no real sound quality. Having used one, great for small desktop speakers, definitely poor for headphones.
FiiO E9, would have been a very good choice. That was around £70 though, and looks like it's been replaced with a more expensive E09K model.