Hot on the heels
In close pursuit of someone/something, or immediately after some event.
This phrase derives from hunting terminology. The 'on the heels' part of this expression is straightforward enough. It just refers to a hunter's literal closeness behind his quarry.
Why 'hot' though? The allusion is to the warmth of an active living creature, as opposed to the cold of a corpse or the 'out cold' of unconsciousness. Trails are said to have gone cold when hounds can no longer sense anything of the prey.
When more excited and animated all mammals become hotter and, when hounds are close to their quarry, they are said to be 'in hot pursuit'. The same allusion is used in guessing games when players are said to get warmer and warmer as they get nearer to success.
This use of 'hot' is quite old and is cited in Milton's The tenure of kings and magistrates, 1649:
"Hungrie Church-wolves following the hot sent of double Livings."
The earliest example of the phrase 'hot on the heels' that I have found is in Edmund Bailey's, History of New Netherland Or, New York Under the Dutch, 1848:
"In Holland, Van de Donck was still hot on the heels of Van Tienhoven."