  We say that a Cohen-Macaulay local ring has finite
$\operatorname{\mathsf{CM}}_+$-representation type if there exist only finitely
many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that
are not locally free on the punctured spectrum. In this paper, we consider
finite $\operatorname{\mathsf{CM}}_+$-representation type from various points
of view, relating it with several conjectures on finite/countable
Cohen-Macaulay representation type. We prove in dimension one that the
Gorenstein local rings of finite $\operatorname{\mathsf{CM}}_+$-representation
type are exactly the local hypersurfaces of countable
$\mathsf{CM}$-representation type, that is, the hypersurfaces of type
$(\mathrm{A}_\infty)$ and $(\mathrm{D}_\infty)$. We also discuss the closedness
and dimension of the singular locus of a Cohen-Macaulay local ring of finite
$\operatorname{\mathsf{CM}}_+$-representation type.
