Ciel, I'd think about trying a new one. I have really, really mild eczema, and my two most common break out areas are my feet and my underarms. It just looks weird to be scratching your underarms all the time.
I don't know that you are similar to me, but the weirdest stuff ended up working. I can use vanicream deodorant and the salt crystal ones, but I don't find them to be effective to me. The one thing that works without reaction for me is Dove Advanced Care cucumber and green tea. Do other dove varieties work? No. Do other brands' cucumber varieties work? Also no.
I hope that you can find your perfect product soon. Underarm eczema massively sucks.