By the time my question went live, I was able to figure it out.  If anyone else needs to do it, the solution is relatively simple.  You simply add another sort node.  Here's an example:  This example sorts by the following fields: Member_Type, State, Institution_Name.  All text fields and all in ascending order.

<xsl:apply-templates select="$currentData" mode="list">
    <xsl:sort select="udt:Member_Type" order="ascending" data-type="text" />
    <xsl:sort select="udt:State" order="ascending" data-type="text" />
    <xsl:sort select="udt:Institution_Name" order="ascending" data-type="text" />
</xsl:apply-templates>

Do you want to change the sorting dynamically?

If not, then you can add multiple <xsl:sort select=".." /> to your script.