Everett received an invitation to the Senior Bowl. He helped the South defeat the North 16-15 and made one catch for six-yards. Unfortunately, he tied Donnel Pumphrey, who stands at 5'8", for the smallest hand of any pass catcher at the Senior Bowl. He attended the NFL Combine and completed all of the required combine drills. Everett also attended South Alabama's Pro Day and opted to re-try the short shuttle and broad jump, but was unable to beat his combine numbers in both categories. NFL draft experts and analysts projected him to be a second or third round pick in the draft. He was ranked the fourth best tight end in the draft by ESPN and NFL media analystMike Mayock, the fifth best tight end by NFLDraftScout.com, and was ranked the sixth best tight end in the draft by Sports Illustrated. Everett was drafted in the second round, by the Los Angeles Rams, in the 2017 NFL Draft. He was the fourth tight end selected in the draft and first player ever to be drafted from South Alabama.
2017 season: Rookie year
On June 13, 2017, the Rams signed Everett to a four-year, $6.04 million contract that includes $3.27 million guaranteed and a signing bonus of $2.53 million. Everett made his NFL debut in the Rams' season-opener against the Indianapolis Colts and caught his first NFL reception on a 39-yard pass from quarterback Jared Goff during a 46–9 victory. The following game, he caught a season-high three receptions for 95 yards in the Rams' 27–20 loss to the Washington Redskins. On October 15, 2017, Everett recorded his first NFL touchdown on a four-yard reception from Goff in a 27-17 road victory against the Jacksonville Jaguars. Everett finished his rookie year with 16 receptions for 244 yards and two touchdowns. He made his playoff debut in the against the Atlanta Falcons. He had a four-yard reception in the 26–13 loss.
2018 season
Everett was a significant contributor in the Rams' narrow 54–51 victory over the Kansas City Chiefs on Monday Night Football in Week 11, catching two touchdowns in the fourth quarter. In the 2018 season, Everett finished with 33 receptions for 320 yards and three touchdowns. In the, he had two receptions for 50 yards in the 26–23 overtime victory over the New Orleans Saints. Everett was targeted once in Super Bowl LIII but had no receptions in the 13–3 loss to the New England Patriots.
2019 season
During Week 5 on Thursday Night Football against the Seattle Seahawks, Everett finished with 136 receiving yards, marking his first 100-plus yard game in his NFL career, but the Rams narrowly lost by a score of 29-30. Overall, he appeared in 13 games in the 2019 season and recorded 37 receptions for 408 receiving yards and two receiving touchdowns.
