Even if they sign Aroldis Chapman, which they want to do, the Yankees are determined to upgrade more than just one spot in their bullpen.

To that end, they remained engaged with Brett Cecil until he signed with the Cardinals, and continue to be among the suitors checking in on Boone Logan, another free-agent lefty reliever.

The Yankees never made a formal offer on Cecil. But that is frequently their style: They monitor an available player, get a sense of what he is looking for, give concepts of what they may be willing to do and then, if they decide the player can be had, try to strike quickly with an aggressive formal offer.

In the case of Cecil, they decided both the dollars and the length of contract were going to be too much. Cecil ultimately signed this week with the Cardinals for four years at $30.5 million.

That contract symbolized that it is a particularly good time to be a free-agent reliever. The postseason accentuated how vital it is to have several quality arms in a bullpen. Cecil was in demand because he is a southpaw who can get both lefties and righties out, and has the ability to induce swings and misses.

The Mariners and Cubs were among the teams that were pursuing Cecil. The Cardinals were apparently aware of the Cubs’ presence, and that might have been part of the reason why St. Louis distinguished itself by going to a fourth year. The Cardinals share a division with the World Series champs, and lost both Jason Heyward and John Lackey to the Cubs last offseason.

The Yankees would love to reunite with Chapman and have him anchor the ninth inning, and have Dellin Betances and Tyler Clippard available to attack earlier in the game. But the Yankees feel they lost a lot of games the past two years because a group of relievers from the minors were either not ready or not qualified to get outs after the starter exited and before the ball was given to Betances, Chapman or Andrew Miller.

They particularly would like to add a reliable lefty setup man, perhaps in tandem with Tommy Layne, who did well last season in his Yankees cameo.