Cheapest and shortest path?
New CDP. Period.
No additional IC to buy no added worries as to the power cord on the DAC either.
My plan? Do Both . IF .
IF you arent spending heavily on the CDP, especially. Transports do matter but you gotta really step up the donation at the Transport store to really see gains there. My former Sony SCD 777 ($3K) and a known as a very good transport if not a great one, was maybe 20% or less better sounding than my $175 Oppo DV 980H, using both a Bel Canto DAC III, and a Lavry D10.
There was about eight years of tech gains across that span inside the Oppo. The Sony was made in 2001. soooooo . Also, my Sony required a motor and laser replacement after five or six years of go to near daily use too.
For DACs. Especially those with a wealth of interfaces, and great jitter reduction significant gains can be had just adding one of them and a cable or two.
DACs are a nice spot to go for immediate improvements in sound quality . IF, again, it has great jitter reduction prowess and if picked out might aid you down the road, should use of a file playback system is desired.
Digital world revolves really fast, So, a new CDP could be the one box solution
But, If DAC, definitely Coax
If nice CDP & DAC . BNC, or AES cables.