One place to find this worked out in detail is the paper "On the Siegel-Tatuzawa theorem" by Jeffrey Hoffstein (published in 1980 in Acta Arithmetica). Lemma 1 of that paper states that if $\chi$ is a quadratic Dirichlet character with conductor $d$ and if $L(s,\chi)$ is nonzero on $(\beta,1)$ and $1-\beta$ is small (specifically $(1 - \beta)^{-1} < 11.657 \log(d)$), then $$ L(1,\chi) > 1.507 (1 - \beta). $$ Also given in Lemma 1 is a lower bound on $L(1,\chi)$ under the assumption that $L(s,\chi)$ doesn't vanish on $(0,1)$.
Combining this with the Dirichlet class number formula $L(1,\chi) = \frac{2 \pi h(d)}{w_{d} \sqrt{d}}$ gives the result you seek with $c_{1} = \frac{1.507}{\pi} c$ provided $c < \frac{1}{11.657}$ and $d > 4$.