Why does the tangential component of the magnetic field increases inside a material?

It sounds to me that you are describing a solid-core transformer. As Matteo correctly explained above the core material is all important in determining the express interactions between the magnetic field and the core.

You mentioned iron, which is a ferromagnetic material. And you mention tiny dipoles, which I assume you intended refer to them as being inside the iron. Remember iron is a base element.

And though there are various theories bantered about, magnetic flux does not necessarily work on a bipolar junction. It is simpler for the human mind to understand magnetic flux as a singular one-way street. Man made magnets themselves display what we refer to as bipolar movement. But that could be just a limitation of our human senses. That man-made magnet, by the way and for the most part is made using the idea that bipolar atoms are realigned in the magnet. And resultingly we see what appears to be two poles. Which is disheartening because it is misleading. Opposite poles appear to be repelling one another. When in fact they are merely interrupting the magnetic flux stream of each other to the extent that the other is pushed away. Like two piglets fighting over the same teet. The phenomenon suggest that the flux attracted to each magnet is so great that it can be considered at maximum flooded capacity. You are pointing two one-way fire hoses toward each other and the pressure is too great to force them completely together. Think about your question in those terms. A magnet is a one way street. And a closed core transformer virtually transforms the primary winding back into magnetic energy whence it originally came at the power station, and the secondary winding then virtually transforms the new magnetic energy back into electrical energy. Because no transfer is perfect, you will never get a larger electrical flow out of a transformer or any metal than what goes in. Magnetic power and electrical power are exactly the same in strength. Our limitations simply force us to measure them differently.