This is a list of the teams in the 2nd round I think that potentially could draft a center. Any team could take a center, no matter what their current situation. The Saints took Cesar Ruiz despite having rookie Erik McCoy from 2019. Nevertheless, these were the teams that seem to have clear needs. It is a long list, I count 13 teams.
Giants (pick 36). Jon Halapio was the starter last year, over Spencer Pulley. Halapio played poorly, with a 56.3 PFF grade, then late in the year had an Achilles injury. Halapio is gone, leaving Pulley on the depth chart. Pulley had a 55.9 PFF grade in 2018, ranking only 31st. Improving the OL is essential to help Saquon Barkley and Daniel Jones.
Panthers (pick 38) As I mentioned in my profile on Matt Hennessy, when he was recruited to Temple, Matt Rhule was the coach there. Now that Rhule is Carolina's new HC, maybe he'll want Hennessy again. The team signed Matt Paradis in FA last year to a 3 year contract, but sometimes a new regime wants to bring in their own guys. Paradis turns 31 in October, has had many injury issues in his career. He was inconsistent last season (63.5 PFF grade). Adding a younger and cheaper rookie could allow them to shed Paradis's salary at some point, maybe even trade him to one of the other teams on this list.
Dolphins (picks 39 and 56) The Fins signed Ted Karras from the Patriots, but this is like signing Austin Blythe (they both got about the same money). Tua Tagovailoa's durability questions are well known. Miami drafted an OT, and adding a center could further help build a line to protect Tua and keep him healthy.
Broncos (pick 46) Denver signed Graham Glasgow in FA from Detroit, but their coach said they see Glasgow as a guard. The Broncos lost Connor McGovern in FA, so adding a center in the draft would make sense to play between Dalton Risner and Glasgow.
Falcons (pick 47) Alex Mack is in the final season of his contract. He turns 34 years old in November. It could be time for Atlanta to groom a new center under him for a year and possibly move on if Mack retires or they don't want to sign him to a new deal.
Steelers (pick 49) Maurkice Pouncey had a 51.5 PFF grade last year, ranking 36th out of 37 centers. Some Steelers fans wonder if he's past his expiration date and needs to be replaced. BJ Finney left in FA, so there is no clear in house successor.
Rams (picks 52 and 57) Sandwiched between two teams who might take a center. But, even if the Rams wanted to move up, how far would they have to go? Would they even be able to make a deal, considering the potential trade partner might also want a center and could be reluctant to do a deal, afraid that the Rams would take the exact player they also want. Brian Allen had a 58.6 PFF grade and had his season end with an injury (MCL).
Eagles (53) Sitting one slot behind the first Rams pick, it could be time to prepare for life after Jason Kelce, who has said that he's year to year at this point and it wasn't even clear whether he would return for 2020. The obvious candidate would be Hennessy, a very similar player to Kelce and who played college ball in Philly at Temple.
Ravens (picks 55 and 60) Against the Rams last year, Matt Skura suffered a very serious injury. Baltimore has enough ammunition to trade up and leapfrog the Rams if they believe they need to and grab a center.
Seahawks (pick 59) Seattle signed BJ Finney as a FA, but based on his history with the Steelers and the salary, he might be used just as a backup. Justin Britt got hurt last year and has been discussed as a potential cap casualty. The Hawks also signed Joey Hunt as a RFA, but he's a backup type.
These are six additional teams that might need centers:
Patriots. David Andrews had blood clots last year and his replacement, Ted Karras, left in free agency.
Chiefs. Austin Reiter is one of the cheap, late round center success stories in the NFL, but he's in the last year of his contract and KC might not want to pay him long term. This would be a good time for them to add a rookie who could take over for him.
Washington. Like Reiter, the Redskins have another solid center who was a late round draft pick, Chase Roullier. He is in the last year of his rookie deal. If they think it might cost too much to keep him, they might want to draft a replacement.
Packers. Corey Linsley is in the final season of his contract. He had a 69.8 PFF grade last season. If GB decide he's not the top FA priority and they can't afford to keep him, drafting a replacement could make sense.
Niners. Paying Weston Richburg big money in FA has failed to solve their center situation and last year Richburg tore his patellar tendon, ending his season.
So, by my count, over half the teams in the NFL have what could be characterized as a pressing need at center.