Am I allowed to change induction hypothesis?

I am doing the exercise from my textbook teaching the induction and stuck for a long time about the induction hypothesis part.

The question is the following:

Let $n \in \mathbb N \backslash \{0\}$, use some form of induction to prove that for all such n there exists an odd natural $m$ and a natural $k$ such that $n = 2^{k}m$.

From my lecture my prof told us that we have to use $P(n)$ to prove $P(k+1)$ and can't use $P(k+1)$ as a precedent, I understand this part, and the following is my approach.

1. Define predicate $P(n) =$ there exists an odd natural $m$ and a natural $k$ such that $n = 2^{k}m$ for all such $n$.




2. Check $P(1)$, if I choose $k = 0$ and $m = 1$ this holds.




3. Assume $P(h)$ is True, namely, I can find such $m$ and $k$ so that $h = 2^{k}m$.




4. Prove $p(h+1)$, I know I have to show two cases since $P(h+1)$, so I assume $P(h) + 1$ is odd first, then I can write $h+1 = 2^{k}m + 1$, and get $h = 2^{k}m$, and by I.H. this is true. (Although I don't think I get this correctly it just looks weird). Then I assume $h+1$ is even, this implies that h must be odd, so induction hypothesis must be changed to $h = 2^{k}m + 1$ in this case, so $h+1 = 2^{k}m + 2$, then get $h + 2^{k}m + 1$, by I.H this is true.

I don't know what goes wrong in my proof, but when I just stare at it I just feel it is not correct, I need help to explain this question, thanks.