The Boston Celtics remained patient this offseason and landed their man, Gordon Hayward, in free agency. They’ll need to make a few more moves to free up enough cap space to make the deal official, at which point the team’s financial flexibility will be all but gone.

That’s perfectly fine. Hayward is an All-Star wing. He will be by far the most impactful player Boston adds this summer—exactly the type of piece you should sacrifice cap space to land.

The Celtics’ roster remains incomplete though. Boston is in desperate need of bigs. They have enough large, switchable wings to lean into a small-ball heavy identity, but even if that is the path forward, the Celtics will need a little more depth down low. If for no other reason, the coaching staff will need to spell Al Horford and provide some injury insurance.

Finding that player may not be critical to Boston’s success, but it won’t be meaningless, and it probably won’t be easy either. The Celtics are very likely to be capped out after Hayward hits their books, leaving them with only the room exception (valued at $4.3 million) and minimum contracts to lure potential free agents.

All of this may change based on trades, but for now, we’ll move forward under the assumption that after the Hayward signing and its associated transactions, the Celtics will still need bigs, and need them on the cheap. We’ve compiled a list of potential candidates below.

The exact profile of the player Boston needs—a big that can rebound, defend the rim, switch pick-and-roll actions credibly, and shoot enough to maintain the Celtics’ spacing on offense—doesn’t exist here. That player is extremely valuable, and has a price tag that far exceeds anything Boston can offer.

Everyone listed below, however, has at least one of the skills Boston needs. We’ve placed them into two tiers: those that can likely be had for the minimum, and those that will probably require the use of the room exception. Everyone is (subjectively) ranked by how ideally each would fit, from best to worst. A few interesting, varied statistics are included for each player, per Basketball Reference.

Tier 1- Room Exception Candidates

This grouping may have been spoiled a bit by the deal the Chicago Bulls gave to Cristiano Felicio. Any agent worth his or her salt would be wise to use the four-year, $32 million deal as a market setter for most of the players included here. Cap space is drying up as free agency marches on, however, and the Celtics may be able to use that to their advantage and land one of these players at a lower rate.

Dewayne Dedmon (UFA): 13.4 RB/36, 1.7 BLK/36 30.6 DRB%

Getting Dedmon with the room exception might be a bit of a stretch. He’s coming off a good year with the Spurs, and it would be surprising if they didn’t have a desire to bring him back. He hasn’t landed a big contract in his career either, so taking a discount to play in Boston is unlikely.

If the market is cool enough on Dedmon for the Celtics to land him, however, he would make for a great fit. He’d instantly become the team’s best rebounder, a gaping hole in need of filling. Dedmon would likely rate out as Boston’s top interior defender as well. San Antonio’s defense was elite with him on the floor and surrendered 5.3 fewer points per 100 possessions when he played.

Dedmon benefitted from playing in the Spurs system, but he has the skills to transfer his impact on defense outside of San Antonio’s robust infrastructure. He even has a little switchability to his game.

Offensively Dedmon is limited. He’s not much of a shooter, but he possesses enough explosiveness to function as a credible roll man, diving to the rim after setting screens. The Celtics wouldn’t need him to be more than that.

Willie Reed: 11.6 RB/36, 1.6 BLK/36, 22 DRB%

Reed is essentially a slightly worse version of Dedmon. He’s not as effective as a rebounder, and doesn’t have quite as much length. Those things are negatives, but they might leave him available at a lower price point, which could benefit Boston. They’d get all the pluses of adding Dedmon but at a slightly diminished rate. That would be plenty for the role they need to fill.

Allan Williams: 14.8 RB/36, 1.6 BLK/36, 31.2 DRB%

Williams is a huge human being and a monster on the glass, but he’s a restricted free agent, which makes pursuing him relatively unpalatable. Williams started games for the Suns down the stretch last year and played well. Phoenix has a glut of bigs on its roster, so they may let him walk if other parties show interest. It seems unlikely, however, that the Suns wouldn’t be willing to match at the level at which Boston could make an offer. It’s probably not worth tying up cap space to test it out.

Kris Humphries: 10.8 RB/36, 23.0 DRB%, .352 3P%

I’m a little shocked I’m including this name, but Humphries can still grab defensive rebounds, and he has, however improbably, turned himself into an average three-point shooter. That’s not a readily available combination, though in Humphries’s case, it comes with enough defensive limitations to make him a last resort. In fact, upon thinking about it, he probably belongs in the next group.

Honorable Mention:

Shouts to a few other potentially interesting, albeit flawed, candidates in Dante Cunningham (too much of a wing), Mike Muscala (not enough rebounding), JaVale McGee (to spacey to maintain Brad Stevens’s sanity), and Jared Sullinger (too many injuries and memories of missed threes). All of these bigs deserve a look if the other players listed here are not available.

Tier 2- Minimum Contracts

Convincing a player to take a minimum contract is tricky business, and it’s possible that the names you see here are looking for something more substantial. Feel free to consider them a part of the lower end of Tier 1 should the market dictate that is where they belong.

Marreese Speights: 24.1 DRB%, 1.1 BLK/36, .372 3P%

Speights is a decent defensive rebounder and a good three-point shooter. He was willing to sign for the minimum with the Clippers, which is a bit surprising, but if he wanted to do the same for the Celtics, it would be well worth a shot.

Thomas Robinson: 14.3 RB/36, 15.8 ORB%, 28.1 DRB%

Robinson is good at two things: rebounding and running the floor. The Celtics wouldn’t be acquiring him to do anything else. He’d essentially be Tyler Zeller without the midrange shots and with an ability to grab rebounds. Robinson might be a bit too one-dimensional for Stevens, but he’s a candidate to fill a role nonetheless.

Jeff Withey: 17.9 DRB%, 2.6 BLK/36

Withey is intriguing as a shot blocker, but his presence in the paint hasn’t actually ever translated to consistent minutes, which is a bit concerning. I’m not a huge believer here, but Withey deserves a mention given his likely availability at a low price.

Andrew Bogut/Festus Ezeli/Tiago Splitter/Donatas Montiejunas/Terrence Jones:

These are all different players with different means of contributing, but they share a commonality: each has, in fairly recent memory, looked like a player deserving of far more than a minimum contract, but due to significant injury, each may be available on the cheap. Boston might be better served rolling the dice and hoping one of these guys outplays a minimum contract (if they are willing to sign for that price) rather than appropriately paying a lower-end talent.