Is There A Word/Idiomatic Expression To Describe The Process Of Someone About To Be Overtaken By A Fellow Rival?

My friend and I watched a stream of the loss of Yankees SP Nestor Cortes yesterday. It was sad. Anyways, my friend mentioned idiom "on the heels of" in our conversation. I don't remember the exact statement he said.
The Free Dictionary gives the following entries for "on the heels of": 1. Following close behind someone, often in an annoying way.I'm a preschool teacher, so I've had toddlers at my heels all day.The interior designer can't get any work done with your puppy at her heels! 2. Close to overtaking a fellow competitor. The leading candidate should be concerned about the underdog at his heels .If she does poorly on this exam, she may lose the title of valedictorian to one of the students at her heels." His usage of the aforementioned idiom got me thinking: are there any words or idioms in the English language that have the same meaning as 2nd sense in the aforementioned entry I listed? I am not asking for the definition of "on the heels of". I am merely asking if there are synonyms for this idiom.