Extract every 2nd row data based value not blank and criteria in column headers

Attached is a sample with desired results. Basically, I am trying to extract the second row that is not blank, based on header criteria in order. If value is not blank, then return the row above as well, and the row header.
Hopefully, someone can point me to a solution.

Time is sorted base on start-time range (9:30,11:30,...)

In C2:

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D2:

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E2:

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Does your example match reality?
can a time be repeated? None are in the example
Is it only sorted on start time? End time has no bearing on the order
In reality how many rows will it apply to?

It matches the reality, but not entirely. It applies to about 100 rows, but is part of a calendar and this is only for one month. Under it I will have October, November and so forth. I will use a macro to change the range of the formula when the current month has passed.
A start time will never be repeated, and the order is based only on start time, end time doesn't matter as they will never be crossed.

@bebo, thank you for your solution, but B2 will be a dropdown and whenever it will be changed, the formula should automatically return the list of times per day and I cannot use a static range as you have here: Data!$H$3:$H$21. Basically if I change the date from 7th of Sep to 25th of Sep, then it should return those time stamps with all the details.

OK, You are right, Data!$H$3:$H$21 range is hard core range, I change it to dynamic range:
INDEX(Data!$B$3:$AE$21,,MATCH($B$2,Data!$B$1:$AE$1,0))
then it should works

See attachment

Please try at
C2:D2

Formula:

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E2
=IFERROR(INDEX(Data!$A:$A,MOD(AGGREGATE(15,6,INT(SUBSTITUTE(LEFT(INDEX(Data!$B$2:$AE$99,,MATCH($B$2,Data!$B$1:$AE$1,)),5),"-",)*24000)*10^4+ROW(Data!$A$2:$A$99),ROWS(C$2:C2)),10^4)),"")

@bebo and @Bo_ry, let me start by thanking you as both solutions work as desired.
I'm hopping that you could help me out with this as well. If my range is calendar based, so I will have the data copied below, and a new row for the days of the month as per the sample, by using the helper column which is the start of the month, would be possible to bring the data by changing the month instead of having to change the ranges in the formula?
Thank you in advance for all the help you can provide

Please try at
C2:D2

Formula:

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E2

Formula:

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Formula can be shorter with volatile offset.

Thank you Bo_ry. Perfection!