To me, the solution is simple - Hatred occurs as normal, counting the turn in which it is "granted" as the "first" round of combat.
I.e. if you fight for 3 rounds without it, then use TEIN to get it, you get the rerolls for one round, and then other Hatred rules (e.g. Must Pursue) for the duration of the combat.
Fluff justification is ridiculously simple. In the middle of a fight, one dude reminds the rest that THESE enemies are the ones they REALLY hate. Fervor ensues, but the adrenaline rush lasts only for so long.
Of course, under RAW the rule is ambigous and needs a FAQ.