The generalized inverse and partial order of tensor are important components of tensor theory.
In this paper, we introduce the T-CS inverse of third-order tensor,
obtain some characterizations and properties of it under the T-product,
and apply it to
introduce a new binary relation:
the ${\textcircled{S}}$ order, which is equivalent to the T-star order under the set of i-EP tensors.
Based on the ${\textcircled{S}}$ order, we further introduce the T-CS partial order and give some characterizations of it.