You should be able to place it on the tool bar using the new "extensions button". Right-click the extension on the button or click the cog alongside the extension and select "Pin to tool bar".

https://support.mozilla.org/en-US/kb/extensions-button

Firefox 109 comes with the Extensions button in the Navigation Toolbar as the new way to manage your extensions. The panel that opens if you click the Extensions button (puzzle piece icon) serves as overflow area for extensions you install. The legacy chevron (>>) overflow area still exists but is only used for built-in toolbar buttons as found in the Customize page.

In the Extensions button panel, you can click the gear icon next to an extension or use the right-click context menu to pin the extension to the toolbar (Pin to Toolbar). To (re)move the extension from the toolbar back to the Extensions button panel, you can use "Unpin from Toolbar" in the right-click context menu.

More Information:

• https://support.mozilla.org/en-US/kb/extensions-button