First please make sure that you have chosen to use the alternate font for the Label Name and not just for the First Line. Setting the alternate font to Label Name will use that font for every name line that is included in the Label Name field in the address.
Next use the controls in the Character Palette to adjust the line spacing for the label name. An easy way to do this using the address template to avoid having to Print Preview is to double-click the address and choose to put the Last Name on its own line. This will use the alternate font on two lines so you can view the line spacing while you make the adjustments.
With certain script fonts you may need to reduce the line spacing significantly. Once you have the line spacing where you want it, the label name lines may partially overlap the address lines in the main font. To fix this, adjust the Baseline Shift for the Label Name alternate font. This will move the entire label name up relative to the address until they no longer overlap (see below).
When you are done, double-click the address again and turn off the option to put the last name on its own line. Then Print Preview to review all your single- and double-line label names and they should be placed properly.