No, you have to designate a new folder when you give the new project a name.
Everyone develops their own method of project organization,
but this has worked for me through several versions of Premiere.
I keep a 'Project Template' folder that contains all of the sub-directories
that I will normally need for each job... Media, Approvals, Documents, Audio,
Rendered Image Sequences, Still Images, Finals etc.
This 'Project Template' folder also contains a template Premiere project that is
pre-set with the interface settings, column layout and project folders I will always use...
Audio (vo, music, sfx), Graphics (sequences, stills, titles, slates),
Cuts (approval versions, finals, and broadcast distribution finals).
When I start a new project I simply duplicate the entire 'Project Template' folder along
with all of its contents, then rename the folder and Premiere project to suit the current job.