  We suggest a new statement of the inverse spectral problem for
Sturm--Liouville-type operators with constant delay. This inverse problem
consists in recovering the coefficient (often referred to as potential) of the
delayed term in the corresponding equation from the spectra of two boundary
value problems with one common boundary condition. However, all studies in this
direction focus on the case of the zero initial function, i.e. they exploit the
assumption that the potential vanishes on the corresponding subinterval. In the
present paper, we waive that assumption in favor of a continuously matching
initial function, which leads to appearing an additional term with frozen
argument in the equation. For the resulting new inverse problem, we pay a
special attention to the situation when one of the spectra is given only
partially. Sufficient conditions and necessary conditions on the corresponding
subspectrum for the unique determination of the potential are obtained, and a
constructive procedure for solving the inverse problem is given. In parallel,
we obtain the characterization of the spectra for the zero initial function and
the Neumann common boundary condition, which is found to include an additional
restriction as compared with the case of the Dirichlet common condition.
