Let \(X\) be a Banach space and let \(B_X\) denote the closed unit ball. In this paper, the authors continue the study of Kottman’s constant \[ K(X) = \sup \{\sigma>0:\exists~(x_n)_{n \in \mathbb N} \in B_X~:\forall~n\neq m,\|x_n-x_m\|\geq \sigma\}. \] They also consider, for closed subspaces \(Y \subset X\), \(s(X) =\mathrm{inf}\{K(Y): Y \subset X\}\). It is known that every infinite-dimensional Banach space has a closed subspace \(Y\) of infinite codimension with \(K(Y) = K(X)\). The authors show that every infinite-dimensional Banach space has an infinite-dimensional subspace \(Y\) with \(s(Y) = K(Y)\). The authors also show that every Banach space is isometric to a hyperplane of a Banach space with Kottman’s constant \(2\). The paper also deals with the corresponding isomorphic versions of some of the concepts.