  We study a dark matter production mechanism based on decays of a messenger
WIMP-like state into a pair of dark matter particles that are self-interacting
via exchange of a light mediator. Its distinctive thermal history allows the
mediator to be stable and therefore avoid strong limits from the cosmic
microwave background and indirect detection. A natural by-product of this
mechanism is a possibility of a late time, i.e., after recombination,
transition to subdominant dark radiation component through three-body and
one-loop decays to states containing the light mediator. We examine to what
extent such a process can help to alleviate the $H_0$ tension. Additionally,
the mechanism can provide a natural way of constructing dark matter models with
ultra-strong self-interactions that may positively affect the supermassive
black hole formation rate. We provide a simple realization of the mechanism in
a Higgs portal dark matter model and find a significant region of the parameter
space that leads to a mild relaxation of the Hubble tension while
simultaneously having the potential of addressing small-scale structure
problems of $\Lambda$CDM.
