Let's take a look at a hypothetical scenario where you might want to make payments, even during the 0% interest period until September 30th. What if you make payments during the 0% interest period, can you still qualify for a 0$ payment or reduced payment at a later time? Paying towards a student loan now will have no bearing on your ability to qualify for a zero dollar payment in the future.
If you have federal student loans, you have the option of making payments through an income-driven repayment plan. Once on one of those plans, you have the ability to change your payment amount at any time based on your income.
So in this example, if you were continue making payments now even though no payments are due, and your interest is 0%, then you lose your job in December, you still have the ability to decrease your payments at that time. The fact that you were able to make payments in the past has no relation to future payment amounts.
Only make payments now during the 0% interest payment if you are planning on paying off your loans entirely (not pursuing student loan forgiveness), have a comfortable emergency savings, have all other debt (with interest) paid off, and do not want to invest in equities. Almost all of these would be better financial decisions in the long run than making payments on a loan with 0% interest, even if it is only temporary.