Looks/sounds like Windows platform .... didn't mention web server ... but *IF* you were on Linux and had full access to httpd.conf file (apache web server config) one could change what apache uses for document root ... on CentOS/LSB Linux flavored boxen that's /var/www/html/
Change that to /var/www/html/moodle/
Restart apache ... then anyone going to http://URL/moodle would get an error but anyone going to http://URL/ would see a whacky Moodle site.
To fix the whacky:
edit config.php file in Moodle code and change the $CFG->wwwroot variable.
and lastly ... since all internal links to Moodle are actually built from DB records:
Do an sql dump of the database and with a text editor search and replace for:
http://site/moodle/ replace with http://site/
Re-import the edited sql file to new/fresh DB for Moodle.
Yes, there is a search and replace tool in Moodle code ... if one can get to it.
Change config.php file again for the new DB name, credentials.
Lastly ... manually remove contents of moodledata/cache/ and moodledata/localcache/ directories ... Moodle caches tons.
And just to make sure everything will come up as expected in case of a reboot/network outage, reboot server. Check server error logs ... especially apache error logs.
Go to site and hopefully one can use the change without any hickups.
Is there any easier way? Well, this is a community ... so let's see if anyone else has a 'shortcut'. ;)
'spirit of sharing', Ken