Depending on your hosting control panel, you should be able to "park" a domain on the existing hosting.
If it's a CPanel server, there'll be an option for "Add-On Domains" - you can add the new domain you've bought to this, and also tell it which folder on the hosting account to use as the "root" of the site.
So, if you had the new site at
http://www.finditcheapest.com/russandmimimakes you would tell the "Add-On Domain" to use "/public_html/russandmimimakes" as the folder.
Does that make sense? If not, happy to help over PM or something if you can't work out which bits to go to in your existing control panels