  We show that if $V$ is a vertex operator algebra such that all the
irreducible ordinary $V$-modules are $C_1$-cofinite and all the
grading-restricted generalized Verma modules for $V$ are of finite length, then
the category of finite length generalized $V$-modules has a braided tensor
category structure.
  By applying the general theorem to the simple affine vertex operator algebra
(resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie
superalgebra) $\mathfrak{g}$ at level $k$ and the category $KL_k(\mathfrak{g})$
of its finite length generalized modules, we discover several families of
$KL_k(\mathfrak{g})$ at non-admissible levels $k$, having braided tensor
category structures. In particular, $KL_k(\mathfrak{g})$ has a braided tensor
category structure if the category of ordinary modules is semisimple or more
generally if the category of ordinary modules is of finite length.
  We also prove the rigidity and determine the fusion rules of some categories
$KL_k(\mathfrak{g})$, including the category $KL_{-1}(\mathfrak{sl}_n)$. Using
these results, we construct a rigid tensor category structure on a full
subcategory of $KL_1(\mathfrak{sl}(n|m))$ consisting of objects with semisimple
Cartan subalgebra actions.
