How do I determine my total active duty time?

Hey Chief, great question. I used the Retirement Services Office (RSO) at my base to get this corrected (since BASD is a major factor in retirement planning). Recommend you visit the RSO at your base and your previous DD 214's (and DD Form 215 if corrections were previously made).

I went through something similar (Army Reserves while in high school delayed entry, then active duty and then ETS'd, college, and then back to active duty). 32 years later I learned that my BASD was never calculated correctly; as such, my DD 214 at retirement would have inaccurate active duty dates.
For example, not factoring in AD dates accrued while a reservist on AD during Basic and AIT. Interestingly, the RSO also told me to visit the Finance Office for a review/audit of my PEBD because it also appeared incorrect; turned out it was wrong and I received a check for the few years that I was underpaid.

Your PEBD should not change.
Your LES should show total years of service, active and reserve.
Your original BASD was the date you first started active service.
Then you had a break in active service during reserve time, so coming back onto active, in order to use BASD to calculate total active service minus the breaks, the BASD would have shifted for the total of the reserve time, minus all days of active duty.
With a discharge of 20080702 and a reentry date of 20090224, assuming no active duty during that, your BASD should have shifted those 237 non-active days from 20060804 to 20070329.