Oh I totally understand the frustration, getting that appointment to line up with your cycle feels like trying to catch a moving target! I went through something similar after my second child, calling the polyclinic a few times before finally getting a slot that matched my period days, so I know how stressful that waiting game can be.
From what I've heard, the reason they prefer to insert during your period (usually around day 3 to 5) is that your cervix is naturally more open then, which makes the procedure easier and less uncomfortable, and it also rules out any chance of an existing pregnancy at that point. My friend went to a polyclinic in the west and said they were quite flexible, accepting her on day 5 when day 3 and 4 slots were full, so it might be worth asking specifically if day 5 is also acceptable rather than only asking about day 4. The slots do fill up fast because the timing window is narrow, so some mums find it helps to call early in the morning right when the polyclinic opens, or even call a few days before your next expected period so you can grab a slot in advance.
That said, do check with your doctor directly about whether insertion outside of your period is possible in your situation, as a specialist would be best placed to advise, hope you get your slot soon, don't give up!
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