This will work but would require a little bit of naming and moving things around to get it done unless you have different clusters to work with. When you run converter against the VM you will have to give it a name. You wont be able to give it the same name but you could make something like computername-tmp. You would basically want to do it in the following order.
1. Use converter to convert the machine into a new machine named computername-tmp.
2. Once that is completed power off your virtual machine.
3. You would then move the virtual disk from the new machine over to your old machine.
4. Remove your old virtual disk for your C drive in the settings and add a new drive pointing to the existing storage and the new C drive virtual disk.
My recommendation would be to just clone the entire machine but that also depends on the size of your data drive. If your data drive is very large then I could understand why you would not want to clone a drive that really doesnt need changes performed on it. If however it is small then i would just save the time and little headache from moving and renaming and just convert the machine into a clone. Rename your old machine first to something like computername-old and then use converter to make a new "clone" of the machine with the updated drive sizes.