Does hydroplaning occur more often in the first few minutes of rainfall due to rain mixing with road oil?

No this is not Hydroplaning, which happens when there is a layer of water on the road that the tires cannot push out of the way fast enough.

Skidding after fresh rain is rather a loss of friction between the road's surface and your tires (due to the road oil). After a few minutes of heavy rain all the oil gets washed off the roads (into the waterways!) and the danger of skidding is lessened.