You don't have to save the document with the merge fields as a template. If you haven't already saved that as a document, do that. The document file already contains all the formatting for the labels, and it also contains a hidden link to the Excel file that you last used to perform a merge.
For the next merge, use the File Open command (Ctrl+O) to open that document again. The first thing you'll see is a message box like this, pointing to the active sheet in that previous Excel file:

If you were to click Yes, you'd get the same data source as before. Instead, click No as Paul's second bullet point said. The document filled with merge fields opens, but there's no data for it now. Go to the Mailings ribbon, click Select Recipients > Use an Existing List, and choose to open the new Excel file.
If the new Excel file has exactly the same field headings in row 1 as the old file, you can go ahead immediately to click either Preview Results to check the correctness, or click Finish & Merge to print the new labels.
If the headings aren't the same, the best way is to change the headings in the new Excel file to match the old ones. Theoretically you could use the Match Fields button, but it will want to change each mismatched field in each label on the sheet -- that gets very old very quickly.
When closing the merge field document, be sure to save it, mainly to save the link to the current data source but also to save any formatting changes you've made since opening it.