The Golomb space $(\mathbb N,\tau)$ (the "universal" counterexample to many questions), also gives a counterexample with $|\mathbb N|=\aleph_0$ and $|\tau|=\mathfrak c$.

I recall that the *Golomb space* is the set $\mathbb N$ of natural numbers endowed with the topology $\tau$ generated by the base consisting of the arithmetic progressions $a+\mathbb N_0b=\{a+nb:n\ge 0\}$ where $a,b$ are relatively prime.

It is well-known that the Golomb space is connected and Hausdorff. Since it contains a countable disjoint family of open sets (like any infinite Hausdorff space), its topology has cardinality $\mathfrak c\le|\tau|\le|\mathcal P(\mathbb N)|=\mathfrak c$.

In place of the Golomb space one can take any other countable Hausdorff connected space.

Such spaces have appeared in other questions of Dominic van der Zypen:

https://mathoverflow.net/questions/300575/is-there-a-connected-t-2-topology-on-mathbbq-that-is-coarser-than-the-euc

https://mathoverflow.net/questions/296581/is-mathbbq-the-continous-image-of-a-golomb-like-space-or-vice-versa

https://mathoverflow.net/questions/280849/cardinality-of-a-set-of-countable-connected-hausdorff-spaces

https://mathoverflow.net/questions/285945/continuous-self-maps-in-the-golomb-space-that-are-neither-increasing-nor-decreas