One does not need a plugin for maintenance mode. Maintenance mode is simply a file in the root named .maintenance. The file should include this PHP at the very top (for no set time):
<?php $upgrading = time(); ?>
and this at the very bottom:
<?php die(); ?>
In between those add valid HTML with what you want in it to the .maintenace file. Test before using on server.
Note: I have had no issues with many sites when this process for upgrade is followed using FTP:
1. Backup site fully and also database.
2. Deactivate all plugins.
3. Switch to (unedited) default theme.
4. Upload .maintenance file.
5. Use FTP to update core WP files, theme, or plugins to latest versions carefully noting any custom content and replacing. NOTE: The folder wp-content contains your themes and plugins, do not replace this folder, just remove specific folders within that are related to your theme or plugins that need updating and upload new versions. Ensure compatability of all plugins with version of WP and your theme first!
6. Remove .maintenance file.
7. Login admin, activate theme and test. Activate plugins ONE at a time, test. Make sure that any plugins that are complimentary to another are activated in proper order.
Has never failed for me.
If failure: Upload .maintenance file right away. Create new db in host and import backup to it. Remove files using FTP (except .maintenance file) and put back old files making sure to update db strings in wp.config.php. Remove .maintenance file. Run site.