You cannot have H1 and TN jobs at same time. It has nothing to do with GC process. You can get anothr H1, or you can 'downgrade' your current job to TN and then get second TN for the new job. If you do the latter, you will need to switch to EAD for both jobs as soon as you file I-485.
So, practically, you cannot get another non-H1 job until you get EAD.