at the heels of (someone or something)

Following close behind someone or something. The cops were at the heels of the bank robbers until they reached the highway. I managed to grab the puppy's leash before he could take off at the heels of our terrified neighbor. I'm a student teacher, so I'm supposed to be at the heels of the English professor all day.

hot on the heels of (someone or something)

1. Following closely behind someone or something. Look, the cops are hot on the heels of the bank robbers! Even though I called for him to come back, Rover remained hot on the heels of the two startled kids. I thought we were hot on their heels, but look—their hideout's empty.

2. Close to defeating or overtaking a competitor. The young politician is hot on the heels of the leading candidate. The underdogs have been hard on the heels of the returning champions for the entire second half. Man, Phillies have been hot on the heels of the Braves in the standings all season.

3. Happening very soon after something else. The new revelation comes hot on the heels of the previous scandal. The team's championship victory was bittersweet, coming hot on the heels of their coach's untimely passing. The announcement of the company's bankruptcy comes hot on the heels of its CEO's sudden resignation just a month prior.

heels of, at/on the

Close behind; closely following. Although these two clichés are very similar, they are not wholly interchangeable. To be at someone’s heels is to be immediately behind, with the implication of chasing or otherwise harrying the person. To be on the heels of someone (or something) means to be following in quick succession (but not necessarily catching or overtaking). Both terms conjure up the idea of a dog being at one’s heels, and both are quite old. John Gower wrote in 1390, “There bene also somme as men sale that folwen Simon ate heles.” And Shakespeare wrote, “One woe doth tread upon another’s heel, so fast they follow” (Hamlet, 4.7).