Change of employer in the middle of the year

I dismissed from my previous job because the client did not renewed their contract with the company. My contract ends aug 26 2015 but before my contracts ends with previous company i was hired by a new employer aug 17 2015. I did a 5hrs shift with my old employer until my contract ends and the company is fully aware. On my backpay i only get the prorated 13th month pay and no tax return. Now in my new employer i had to pay more for the tax or i still have tax due. My question is did my previous employer remit the tax that was deducted to me or were they able to remit those taxes? People are saying that they will not be able to remit those taxes since i resigned in the middle of the calendar year. But they were able to provide 2316 to me. Please advise.

If you have a BIR form 2316 that is all you need.

It's not your place to question if the previous employer remitted the tax. And yes, the company is required to remit taxes monthly - so it can be assumed that all taxes were paid to the government.