I had heard that it wasn't possible to change this setting, so I informed my co-worker, unfortunately, we can't do anything about it.
But as usual, I hate giving the "It's not possible" answer, so I started to dig into the issue and see if someone had a solution. Lo and behold, someone had! Thanks to Aveenav who posted the answer on SharePoint StackExchange.
If you're familiar with SharePoint 2013, implementing the solution is relatively straightforward. However, I've included instructions below to help the rest of us out.
1. Go to the list or library where you want to see additional views.
2. In the upper right corner of the page, click the cog and select Edit Page.
3. Click Add a Web Part in the main window area.
5. In the Script Editor, click Edit Snippet.
<script type="text/javascript">
ExecuteOrDelayUntilScriptLoaded(overrideSurfacePivotCount, 'clienttemplates.js');
function overrideSurfacePivotCount() {
ClientPivotControl.prototype.SurfacedPivotCount = 4;
};
</script>
7. The "4" in the script above determines how many view selections to show. Change this number as needed. Click Insert on the Embed window.
8. Click Stop Editing in the upper left of the SharePoint ribbon.
9. Test the views.
I've increased the number of view selections available to 15 and encountered no issues. If the views are extended to the point of going beyond the width of the page, the view selections will wrap to the next line.
Hopefully, this procedure will be handy for you if you get a similar request.