ben,
Getting that first job as a fresh CPL is a tough one. From my observations in FAA land, instructing (with another expensive rating required) is the most common job for a new CPL. But, I don't know if there is any shortage of CFIs lately to help qualifying for an E3. There are surely lots of new people here asking the same question you have.
Non-instructing jobs will commonly fall under FAR Part 135 and will require about 500 hours TT for a VFR-only spot...go up from there...and those are likely few and far between:
https://www.ecfr.gov/current/title-1...-135/subpart-E
I hope someone will come along shortly with more info. Good luck and hang in there !