Apply to EVERYONE! If they operate an aeroplane bloody phone & write to them! Jet, TP, piston, single, airline, air taxi, traffic spotting, sky diving, glider tug or even hang around an aerodrome getting to know those who operate the things. Be around when someone needs a hand to push his/her aeroplane or would appreciate someone showing them where the aerodrome cafe is.
But be realistic. You WILL get many, many, many (repeat ad nauseum) knockbacks. You only need ONE acceptance.
As for the "I know/don't know someone/anyone who did/didn't get a job on a jet/TP/piston straight from training/after years" debate:
A very, very FEW lucky individuals gain employment on a jet/TP straight from training. By this I mean without an intermediate job or series of jobs. It could be weeks or is more likely to be months/years. 'More likely' refers to the VERY FEW who are fortunate this way.
Most, however, face traffic spotting, glider towing, skydive dropping, occasional ferry (to the next aerodrome where the maintenance is done, not US to uK) & - even more mostly - instructing.
Many will never get a jet airline job. Many will finish their careers on TP. Many others on pistons.
For most people getting your licence is the easiest part of working as a pilot. Getting the first job and every single subsequent career advancement is far, far more difficult.