Create New Tab Sessions with the same working Directory
One of the options for a profile is which directory to use when creating a session with that profile. The default is to use your home directory, but you can change that to use the previous directory instead. You can change this setting in the default profile (allowing you to continue using ⌘-T), or you can create a new profile which uses this setting, and assign a different keyboard shortcut to create a session with that profile.
Preferences->Profiles->Working Directory->Reuse previous session's directory