First, if this is a production system you really need to be running this as a replica set. Running production databases on singleton mongodb instances is not a best practice. I would consider 2 full members plus 1 arbiter the minimum production set up.
If you want to go the replica set route, you can first convert this instance to a replica set:
http://docs.mongodb.org/manual/tutorial/convert-standalone-to-replica-set/
this should have minimal down time.
Then add 2 new instances with the correct storage set up. After they sync you will have a full 3 member set. You can then fail over to one of the new instances. Remove this bad instance from the replica set. Finally I'd add an arbiter instance to get you back up to 3 members of the replica set while keeping costs down.
If on the other hand you do not want to run as a replica set, I'd shutdown mongod on this instance, copy the files over to the new directory structure on another appropriate volume, change the config to point to it (either changing dbpath or using a symlink) and then startup again. Downtime will be largely a factor of the size of your existing database, so the sooner you do this the better.
However - I will stress this again - if you are looking for little to no down time with mongoDB you need to use a replica set.