pingpong02,
Definitely it is easier to get a job with GC rather than H1. If it is H1, new employer need to spend time and money sponser for H1 and people will be a bit worried to do that initial investment thinking "What if it is not a better fit"? If it is GC it is simply "hire-and-fire" not much risk involved. Knowing this fact it self means a lot in terms of reducing level of stress.
I am not able to anwser second part of your question because I didn't try to look for a new job (actually in past 4 years I didn't see my resume at all)!
PCee