I have answered my question at SE. Of course, any comments and answers are still welcome.

I think I have a misunderstanding for the KK theory. In the KK theory, we are living in, say, a 5-dimensional spacetime with one dimension compactified. What's different from the brane-world theory is that, in brane-world theory, we are living on a 4-dimensional brane which is embedded in the 5-dimensional spacetime. So in the post, I can not assume that the extended world is at $y=y_0$. Actually, in KK theory, the particles can be in principle everywhere. There is no a 4-dimensional subset which can be identified with our observed 4-dimensional spacetime. But since we observe the world by exchanging momenta and energy with the objects and also the compactified dimension is very small, all the low energy particles are frozen in the extra small dimension so that there is no exchanges of momenta in the extra dimension. In that case, the particles can not feel the existence of the small extra dimension.

Note since the small extra dimension is compactified, the minimal momentum for a moving particle in the extra dimension can be obtained according

$$e^{ipL}\rightarrow p=\frac{2\pi n}{L}.$$

So if $L$ is very small, the first excitation energy $\frac{2\pi}{L}$ to move the particle in the extra dimension is very large. Equivalently, the low energy particles are frozen at that direction and can not feel the extra dimension. In a word, the momentum excitation is gapless in extended dimensional space while not in compactified dimensional space.

This case changes in string theory. Beside the momentum along the extra small dimension, the string can wind around the compacted dimension, which becomes a quantum quantity after quantization. When the extra dimension becomes smaller and smaller, the excitation spectrum for the winding number becomes continuous. The gapless excitations emerge again. In that sense, the extra dimension actually are becoming bigger again.