How to have a 2-color logo put on a color background of the same hue?

The logo I currently have is similar to the one below.

It has 2 colors of the same hue, with different brightnesses. I'm attempting to put the logo on a Windows app, with the design calling for a colored background and a white icon. I don't want it to have a white background and look out of place on any user's Start screen, however, I'm wondering how I should change the secondary color in the logo. I don't know if it should look something like this,

or something else. What's the most visually appealing way to do this?

Answer

I actually feel your current iteration may be the best solution.

Out of the possible variations I explored the white and secondary color version was the best in terms of clarity and prominence, in my opinion.

Samples....