In this paper (also available here) we show that a graph having no odd cycles of length $\leq 2r-1$ and having at most
$$
\frac{(k+r)(k+r+1)\dots(k+2r-1)}{2^{r-1}r^r}
$$
vertices is properly $k$-colorable. This provides an upper bound.

Another upper bound is provided by the paper by Kierstead, Szemeredi, and Trotter (see reference [4] in the above paper --- this also contains some upper bounds!).

On the upper bounds --- there are several different examples. One is the Schrijver subgraph of the Kneser graph (see [6] in the mentioned paper, or discussion on the first page), another one is the iterated (generalized) Mycielski construction applied to $C_{2r+1}$; see a relate discussion in

A. Gyarfas, T. Jensen, and M. Stiebitz, On graphs with strongly independent color-classes. // Journal of Graph Theory 46 (2004), 1–14.