Pass degree holder here, was in similar dilemma when I first started out. Situation was a bit more bleak as I wasn't interested in my field of study and it was 2009, financial crisis.

Did some freelancing work while sending out resumes. Eventually landed a job in a small firm doing project mgmt. I feel if you are just starting out, try going for smaller firms if you are trying to land a job first. No point try applying to big firms and compete with other people better positioned. Gain some experience first before trying bigger firms. Once you get pass the fresh grad hire phase, it becomes a bit easier.

I disagree with the stereotyping for GLCs and SBs though, been to a few for interviews before, and got shortlisted before. For fresh grads, grades is probably a big consideration for them. But what they are really interested in is whether you are familiar with your work and whether you can perform. This is especially for technical jobs; engineering, project mgmt, IT etc. So gaining a few years of experience is good if that's your cup of tea.