very close behind sb / sth; very soon after sth; following sb / sth very closely
(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.
hot on sb's / sth's heels
following sb / sth very closely
He turned and fled with Peter hot on his heels.
Definitions from the Oxford Advanced Learner's Dictionary, 9th edition