A lot of the antiperspirants have the same main ingredient. One that I commonly see is Aluminum Zirconium. It's quite possible and likely that the Secret deodorant, which seems to be working for you, actually has the same main ingredient as those others that your body rejects. The problem then probably lies in the fragrances.
I happen to have sensitive skin and very sensitive sense of smell. There are very few antiperspirants / deodorants that I can use because the smell of most of them is too strong for me. I'll either get headaches and / or be sneezing.
You should try either unscented antiperspirants ... and make sure they're "antiperspirants" vs. simply "deodorants" or antiperspirants with the mildest fragrance you can find.
In conjunction with this, if you haven't done so, you can try antibacterial soaps like Safeguard or Dial. I use Safeguard myself. Dial leaves too much of a white film on your skin. But, Safeguard is OK. An antibacterial soap should help reduce or maybe even eliminate the BO problem.