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Answering a question of Wright, we show that spheres of any radius are alwaysconnected in the curve graph of surfaces $\Sigma_{2,0}, \Sigma_{1,3},$ and$\Sigma_{0,6}$, and the union of two consecutive spheres is always connectedfor $\Sigma_{0, 5}$ and $\Sigma_{1,2}$. We also classify the connectedcomponents of spheres of radius 2 in the curve graph of $\Sigma_{0, 5}$ and$\Sigma_{1,2}$.

Optimal connectivity results for spheres in the curve graph of low and medium complexity surfaces