Is there a limit ordinal $\kappa_0$ such that $\aleph_0 < \kappa_0 \leq {\frak c} = 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\leq 2^{\aleph_0}$ there is a connected $T_2$-space $X_\lambda$ with the following property?

$\lambda$ is the smallest ordinal such that $X_\lambda$ contains no subset isomorphic to $\lambda$ with the order topology

Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\lambda$ with the following property?

$\lambda$ is the smallest ordinal such that $X_\lambda$ contains no subset isomorphic to $\lambda$ with the order topology