  We continue the effort of grokking the structure of power-bounded $T$-convex
valued fields, whose theory is in general referred to as TCVF. In the present
paper our focus is on certain expansion of it that is equipped with a tempered
exponential function beyond the valuation ring. In order to construct such a
tempered exponential function, the signed value group is also converted into a
model of $T$ plus exponentiation and is in fact identified with (a section of)
the residue field via the composition of a diagonal cross-section and an
angular component map. In a sense, the resulting universal theory TKVF is a
halfway point between power-bounded TCVF and exponential TCVF. This theory is
reasonably well-behaved. In particular, we show that it admits quantifier
elimination in a natural language, a notion of dimension, a generalized Euler
characteristic, etc.
