OP - you have an excellent DAC. I don't think you'll find anything dramatically better. I do think the AN DACs sound pretty different.
Something in the ballpark that is modern and very affordable is the MDHT Orchid. NOS, R2R, tube buffered output stage, async USB input up to 24/192. Mine is modified with a crown chip, upgraded caps and resistors, and some massive Duelund coupling caps that cost a fortune.
Back in '98 I had a DAC3 Signature (OEM) while working at an Audio Note dealership. It was really great and a world away from the Mark Levinson gear that was top of the class in digital at the time. My amps were single ended 300B I had built with a friend. Transport was Sony's top of the line ES player, which was a terrific piece of equipment I could hardly believe was Sony. Cable was hilarious Yamamoto the size of a fire hose waaay before cables of this size became common.
I extensively modified the power supply of the DAC3 with a separate tube rectified power supply that was larger and weighed more than the DAC3. I also experimented with different tubes and coupling caps. Took it around to a few other stores and high end systems of customers and people were totally blown away. Liquid, inner detail, masterful imaging and staging, and the pacing was terrific. Not as detailed some might say.
To me with AN DACs you're mostly hearing the output stage, and so if you like their amps and understand what they do for sound, the DACs bring that to any system.
If you're into tube rolling or modifying gear, the AN chassis is incredibly user friendly. No wonder they use it for the kits.