The short answer is no - no free skyclub admission with any domestic ticket. First class or Coach.
The exception to this is the premium Delta 1 Transcon product Delta offers from JFK to SFO and LAX, but these are the exceptions, not the norm.
If you want a free way into the Skyclub the only choice is fly 100k miles a year and earn Diamond Medallion, open a credit card like the AMEX Delta Reserve card or the AMEX Platinum Card, but both carry a $400+ Annual Fee or buy a day pass, which costs ~$50 (although a discount is available if you have the AMEX Delta Gold Card).