Don't allow tantrums to start ever. Once she has the self control and understands it's not allowed, she will be able to control it when you explain something to her.
My daughter (4) is super sensitive by nature, but because she has never been allowed to have tantrums, when I see the eyes welling up I can give her a calm, "No crying, babe, calm down and tell me what you want to say" and she stops herself right away and we can deal with the issue at hand. I still listen to her express herself and comfort her, but she is not allowed to begin a fit, and she's never carried one through (except the first couple of times she tried them before we nipped it when she was 18 months old-and then she tried a couple in public at 2 1/2, again-NIPPED in no uncertain terms)
You do have to use firm discipline (time outs don't work-they just tantrum in them) to train against this, which many people can't bear to do (which is why ignoring and closing kids off in their rooms to freak out alone has become acceptable these days), but it enables your child to hone their self control quickly and not let things get out of control in situations like the one you're describing.
Teach her not to act out and go into a frenzy. She needs that skill. Start at home and eventually she will follow your direction at drop off too.
At the very first couple of sobs before it escalates: One warning, and firm consequence if she chooses to continue. Every time.
At three, you can explain exactly what the new rule is about fits, what the result will be, and she'll understand. You'll still have to follow through because explaining won't be enough to nip the habit, but she'll learn quickly since she's old enough to understand. Once she sees you mean it, she'll stop. You won't believe she ever did freak out.
If you train her not to have fits, it won't matter what the situation is and explaining will be all you need. Until she's disciplined, explaining won't help-you'll just have to wait until she feels like outgrowing it.
It's easy to cloud the waters by attaching this to a good- bye making it a legitimate separation anxiety thingy, but I'm telling you, kids who don't tantrum, don't tantrum. Even at good byes. I may have been 'mean' to some up front about the behavior, but my daughter cheerfully hugs and kisses me good bye for school-always has, and she expresses her emotions with words and controlled sadness. If she's crying, I know it's legit, because she never has the self induced frenzies. You can do it,good luck, I know it's hard!