digiCamControl

Currently digiCamControl v2.0.65 can generate file names based on file exif data, but these seem to correspond to photo end time, which differs to start time for long exposures.

exiftool.exe suggests that start time fields exist and even to 1/100 second precision (using Canon 6D).
digiCamControl may be using "File Modification Date/Time", but it could use "Create Date", "Date/Time Original" or "Modify Date".
Note: the latter 3 appear twice in exiftool.exe output, first without milliseconds rounded down to nearest seconds, then with milliseconds. Reading the occurrence with milliseconds (if available) and doing a normal round woulds be nice and or another filename TAG could be introduced to preserve the milliseconds.

This suggestion may not be practical because exiftool.exe may be needed and it can be slow to run, unless it is opened once and kept open in the background. Also exif data may be different for other cameras. Also "File Modification Date/Time" may be a PC time where as the others may be a camera time, not sure if there are times when PC time is desirable.

I already use a application to read exif data so this not a problem, can you find a exif tag (should be a hexa numeric value ) where this start capture time is stored ?

For me exiftool (v10.23) reports that 0x0132, 0x9003 and 0x9004 are always same and I believe corresponds to the start time rounded down to the nearest seconds. Unfortunately exiftool does not give a hex code for the second occurrence of these three fields which has the same time to nearest 1/100 second. Perhaps your exif reader will simply give milliseconds using any of the above three codes..

Here are some of the fields from the attached exiftool output: