Will A new DAC improve the sound of my Cary 306/200 CD Player


I have a Cary 306/200 CD Player which I purchased in 2003 for about $5000.00.
I am happy with said player, but would it make sense, since DACS have vastly improved over the past few years to
bypass the current DAC and purchase a DAC, in the $1000.00 range, such as the Mytec Liberty Dac to improve the sound. I am happy with the transport and my rationale is that for $1000.00 or so, I can essentially greatly improve my sound for a modest investment. Appreciate any recommendations, as well as any thoughts regarding my reasoning. The
question is would bypassing the old DAC with something up to date, make a big difference therefore justifying the
expense?
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I would say it's very likely the sound will be different, and it's up to you to decide if it's an improvement or not :) That said, if you can demo anything, or if anyone does like Schiit where you can audition for up to 15 days and send back for full refund (minus shipping, unfortunately - cost of doing business I guess), that's what I would suggest. Musicdirect, Amazon, and other places can do so, and a local dealer might as well, but just be clear on the policy if you attempt to do so.

It's a little over your price range but the newest Gungnir from Schiit reads like a terrific value for around $1250. I have it's bigger sibling, Yggdrasil, and it makes redbook CD sound absolutely glorious. 
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I have the same CD player as you and over the summer bought a gently used PS Audio PerfectWave DAC for about a couple of hundred dollars more from your price range.  The synergy sounds great to my ears.
I'd recommend trying an Audio Alchemy DDP-1 with the PS5 power supply. I've been running this combination for about a year & it is remarkable for the money. I placed a link below in case you're interested.

https://www.audioadvisor.com/prodinfo.asp?number=1ALDDP1P5

 
Note that you are not listening to the DAC, but the output stage. You may be able to find a more modern DAC, but you will have to pay a lot for an output stage which betters the one in your CARY.