What I currently do is I link the project to another location and give it the same name, then check the box where it says \"Branch after share.\" And then I would Check out
You can find a good reference here: http://www.codepool.biz/version-control/sourcesafe/branch-in-sourcesafe-vss.html
Basically right-click-drag your folder to where you want a branch, and when you let go you are given share/branch/recursive options.
Shudder.
I think the way you describe in the question is the only way you can do it in sourceSafe.
I usually name the copied directory "V1.0" (or whatever is appropriate) and keep them all in a folder that is the main project name.
That is the generally accepted way of branching your source code in SourceSafe. The only other way to do it, if merging and retaining the history are not an issue, is to copy the files to a new folder, remove the read-only attribute, remove the .vssscc and .scc files, and then add that new project to SourceSafe. At that point, you have an all new project, with no prior history.
The way you described is the only supported way to do "branching". And as you pointed out it is rather clunky. In VSS it's best to avoid branching alltogether as it will destroy your source history.