What the guys above said. However, word of caution - this knocking on doors malarky never worked for me, be aware it can backfire and get you noticed for all the wrong reasons.Especially if the relevant department is busy and doesn't want interruptions.
Sad truth is, nobody in the airlines really gives a toss about SE piston time.It's an irrelevance to them. Some of the forms I have been filling in for foreign airlines didn't even have space to fill it in anywhere....it was 'total jet hours', 'total jet command', 'total jet command on type'. I lost my 1800 odd hours on cessnas on several occasions.
Corporate operators may look unfavourably on your 737 rating.Cos if that's on your licence they know where your heart lies - airlines.
Best of luck, it's tough out there.