I'd say asking after each deadline is fine (as others mentioned, even good, since it shows interest in the job). Unless headhunted, I would go the route of
- See notice about job
- Read up on organization (preferably about the location the job is)
- Call listed people, ask about the job, how many applicants so far, etc
- Apply for job with mention of your call. This will immediately make you "oh yeah, that person" Instead of "random applicant #142"
- Call a week later to make sure they got the application and all attachments. They will then look through applications, again, taking notice to yours. This way, you are already not in the bottom of the pile so to speak. When they get to it, you will already have registered with them, so when it becomes time to weed down applications for interviews, you will almost always be picked over the people who have just filed an application without taking any contact/interest above that. It's crazy how many people do not take contact with the place they are applying for a job at any point during the application process.
Then there is the point you are at. Calling again, which should be fine to do after each deadline has passed.
Be careful with mentioning instability / uncertainty of your current job though, as might impact negatively on your negotiations of a paycheck if you are offered the job.
That said, stuff happened quickly for me now. Sent an e-mail on Friday to the leaders of an organization if I could send them an open application (met them before). 1 hour later, got an e-mail back asking me to come to an interview today (didn't have to send any application or references). Met at interview, before I got my coat off, head of organization said "Yeah, we want you. You got the job, now excuse me, I need to go take a smoke". Higher pay than my current job too. Yay for the efficiency of private organizations!
