  We investigate an invariant for continuous fields of the Cuntz algebra
$\mathcal{O}_{n+1}$ introduced in our previous work, and find a way to obtain a
continuous field of $\mathbb{M}_n(\mathcal{O}_\infty)$ from that of
$\mathcal{O}_{n+1}$ using the construction of the invariant. By Brown's
representability theorem, this gives a bijection from the set of the
isomorphism classes of continuous fields of $\mathcal{O}_{n+1}$ to those of
$\mathbb{M}_n(\mathcal{O}_\infty)$. As a consequence, we obtain a new proof for
M. Dadarlat's classification result of continuous fields of $\mathcal{O}_{n+1}$
arising from vector bundles, which corresponds to those of
$\mathbb{M}_n(\mathcal{O}_\infty)$ stably isomorphic to the trivial field.
