Chern isomorphism in odd theory

Question

Let $M$ be a manifold (closed). I know the following result connecting K-theory with cohomology: there is a natural isomorphism $K^0(M) \otimes \mathbb{C} \cong H^{ev}(M,\mathbb{C})$ (the same is true for the field of rationas and real numbers) where $H^{ev}$ is the even cohomology ring. I would like to deduce the same statement for odd theory, namely that $K^1(M) \otimes \mathbb{C} \cong H^{odd}(M,\mathbb{C})$. The idea is to use suspension $SM$ and already established isomorphism (for example it is true that $H_{n+1}(M) \cong H_n(X)$ for $n \geq 1$ and $H_0(X) \cong H_1(SX) \oplus \mathbb{C}$ and $H_0(SX) \cong \mathbb{C}$): I'm pretty sure that this is everything which is needed. However I'm getting some (single) extra copy of $\mathbb{C}$ and I'm wondering whether there are some nuances concerning reduced versions of the theory etc.