Sony XA5400Es CD Player DAC Upgrade?

I was wondering if using an outboard DAC with my Sony CD player would improve the sound?

It sounds good now, but I am curious if anyone has used other DACs to improve the sound. 

I use it in a bedroom headphone system. The music goes from the Sony player to a Woo WA22 headphone amp.
Yeah, I hate to say it but....  I get better (if not Whole World of Difference) fidelity in every parameter when I route my XA5400's Red Book output through my much-maligned-on-this-website's Mytek Brooklyn Bridge.  With hybrid discs, the Mytek BB actually makes my Sony's Red Book CD output come whisper close to what I get from the Sony's SACD feed. Yeah, it could be an EQ difference, but sometimes I actually prefer what the Mytek delivers.  
I haven't looked into Mytek yet. have several SACDs but listen mostly to Redbook. 

BTW, if I add a DAC the price range would be south of $1,000.(used or new)
I've owned the 5400 and the CD was not bad, but any of the $1000 dacs mentioned in the forum threads would by far sound better than the 5400.  Whether you can hear the difference is another story.  That's not a jab, btw.  People listen for different qualities and in some cases, moving to a better dac doesn't make a difference to what they're listening for. If you want a recommendation, I think the Heed Abacus is excellent.  Well-made, American distribution and service and sounds great, IMO.  A bit over your price point, but just a bit. 
Why not try the Khadas Tone Pro DAC? $190! It measures near- state-of-the-art! I have the earlier - and cheaper - version. I personally think that four-figure DACs are over-rated and are examples of confirmation bias!
One thing I should mention is that I use balanced XLR to the Woo amp.

I would  also like to have a headphone output to contrast SS vs Tube.