With Cousins in tow, the Pelicans worked with Jones' representatives to find a more favorable situation, but no trade materialized ahead of Thursday's 3 p.m. ET deadline.
Now that he has cleared waivers, Jones, 25, will get to seek opportunities elsewhere.
"We released him because, obviously if you look at the big [man] situation now, there was not going to be a whole lot of minutes, I think, for him," Pelicans coach Alvin Gentry said. "He wants to play. Obviously he's a guy in a contract year. We just felt it's best to let him go."
ESPN's Chris Haynes first reported Sunday that the Pelicans were seeking a trade partner for Jones to ease a frontcourt logjam created by the addition of Cousins.
Hollis Thompson, a swingman who spent three-plus seasons with the Philadelphia 76ers, signed a 10-day contract with New Orleans on Thursday.
The Pelicans will now have one open spot on their 15-man roster.
ESPN's Marc Stein reported Thursday that New Orleans will audition veteran point guard Jarrett Jack in its quest to shore up its backcourt. Jack, who spent 95 games with the then-New Orleans Hornets between 2010 and 2012, hasn't played in the NBA since tearing the ACL in his right knee in January 2016.
Jones, a teammate of Davis' at Kentucky, signed a one-year deal with New Orleans this offseason for the veteran's minimum after four seasons with the Houston Rockets. He is averaging 11.5 points and 5.9 rebounds in 51 games, including 12 starts.
New Orleans is 23-34 overall and 2½ games behind the Denver Nuggets for the eighth and final playoff seed in the Western Conference.