very close behind sb / sth; very soon after sth; following closely behind sb
(hard / hot) on sb's / sth's heels
very close behind sb / sth; very soon after sth
News of rising unemployment followed hard on the heels of falling export figures.
at / on sb's heels
following closely behind sb
He fled from the stadium with the police at his heels.
Definitions from the Oxford Advanced Learner's Dictionary, 9th edition