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Americans often use after instead of past in telling the time, for example, ten after six.

But do Americans say half after ... in telling the time?
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Oddly, we don't say "half after" but we do say "quarter after" and "quater of" or "quater to." (The latter interchangeable spoken, but "quarter to" in writing."

"Half past" isn't used very often.

I know it's very idiomatic, but you'd be more likely to hear "I'll be there about quarter after, maybe 6:30 if the traffic is bad." Or even "I'll be there about 6:30, maybe quarter of, if the traffic is bad." In a situation like that, where it's clear from context that you mean "quarter to 7:00," you'd leave that off in speech."
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Good examples GG. I never thought about those much, but being idiomatic certainly plays a big role!
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Comments  
No. Half past six or six-thirty are the only normal options.
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Dear Mrs. Kindhart (or Martha):
Will you lunch with me on Monday the tenth at half after one o’clock?
Hoping so much to see you,

Sincerely (or affectionately),
Jane Toplofty.
Emily Post (1873–1960). Etiquette. 1922.