You need to duplicate your theme to the second store you want to have design change.
Example, if the change only apply to CSS and images, you skin folder will have
second store (duplicated from your-main-custom-theme. The CSS and images folder should contain ONLY the files for the second store/
Suppose you only need one image change for second store, say, logo.png, then ‘logo.png’ is the only image file that goes into the “images” folder.
Do similar directory structure for “design/frontend/default/your-main-custom-theme”
You also need to setup the “your-second-store’ theme in “system/configuration/Design/Themes” under your “second-store” config, NOT the Default Config.