Can I retrieve a re-hashed (hash) value, for instance by additional hashing? I have a double-SHA-256 of some text $$ h_2 = \operatorname{SHA256}(\operatorname{SHA256}(m)).$$ I don't need the plain text ($m$), but somehow I need to get first $$h_1 = \operatorname{SHA256}(m).$$

Can I have it ($h_1$) by keep hashing the second one ($h_2$)?
$$ h_1 = \operatorname{SHA256}(\operatorname{SHA256}(...\operatorname{SHA256}(h_2)...)).$$
Short answer; Almost NO!

Can I have it by keep hashing the second one?

There were great answers for the question; Cycles in SHA256
In short, if we model SHA256 as a uniform random function then the probability of element being on the cycle is
$$\frac{1}{\sqrt{\hspace{.03 in}2\hspace{-0.05 in}\cdot \hspace{-0.04 in}\pi} \cdot 2^{127}}$$
The average cycle length with expected value for SHA256 is $$2^{127} \sqrt{2\pi}$$
Simply consider the first hash $\operatorname{SHA256}(m)$ as the starting point;

As one can see that being on a cycle has a very low probability and you almost find none in a cycle since the probability is $\frac{1}{\sqrt{2\cdot \pi} \cdot 2^{127}}$.
Even it is on a cycle, you almost certainly can not calculate the cycle to find the pre-image since the average cycle length is $2^{127} \sqrt{2\pi}$.

These are from the result of Bernard Harris's magnificent work; Probability Distributions Related to Random Mappings in 1960.

Also, this exists in the Handbook of Applied Cryptography - Fact 2.37

tail lenght = $\sqrt{\pi n /8}$
cycle lenght = $\sqrt{\pi n /8}$
rho-lenght = $\sqrt{\pi n /2}$