Hawking radiation is always present around a dynamical black hole -- it is produced by the dynamical spacetime itself[1]. (All black holes that aren't eternal -- that includes any that form by gravitational collapse of matter -- are dynamical, and thus have Hawking radiation, even while they're growing.)
In principle we should be able to detect Hawking radiation as black holes first form, since it will backreact with the black hole, and probably even interact with the collapsing matter. Studying BH-forming supernovae and the like will lead to discoveries in this difficult area of https://en.wikipedia.org/wiki/Semiclassical_gravity as hot Hawking radiation is in principle directly observable, and there will be indirect traces.
The problem is that BH formation typically happens in a bright environment. The candidate black holes we know about aren't that young and as a result the Hawking gas will be cold enough to have negligible impact: basically no interaction with nearby matter, basically no backreaction on the black hole itself, and much colder than the surrounding environment (infalling matter including the CMB gas) and thus in practice impossible to detect directly with telescopes.
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[1] Well, more precisely, given Einstein-Maxwell electrovacuum and general relativity with a black hole metric, Hawking radiation is inevitable. Hawking's original work dealt with a static spacetime (i.e., an eternal, unchanging black hole) and used negative energy quanta as a trick to proxy for a dynamic spacetime. Using a dynamical black hole (i.e., one that grows and shrinks), one does not need negative energy quanta at all, much less a mechanism which tosses only those halves of pairs into the BH (in order to keep the metric unchanged from pure static Schwarzschild).
