  The lace expansion for the Ising two-point function was successfully derived
in Sakai (Commun. Math. Phys., 272 (2007): 283--344). It is an identity that
involves an alternating series of the lace-expansion coefficients. In the same
paper, we claimed that the expansion coefficients obey certain diagrammatic
bounds which imply faster $x$-space decay (as the two-point function cubed)
above the critical dimension $d_c$ ($=4$ for finite-variance models), if the
spin-spin coupling is ferromagnetic, translation-invariant, summable and
symmetric with respect to the underlying lattice symmetries. However, we
recently found a flaw in the proof of Lemma 4.2 in Sakai (2007), a key lemma to
the aforementioned diagrammatic bounds.
  In this paper, we no longer use the problematic Lemma 4.2 of Sakai (2007),
and prove new diagrammatic bounds on the expansion coefficients that are
slightly more complicated than those in Proposition 4.1 of Sakai (2007) but
nonetheless obey the same fast decay above the critical dimension $d_c$.
Consequently, the lace-expansion results for the Ising and $\varphi^4$ models
so far are all saved. The proof is based on the random-current representation
and its source-switching technique of Griffiths, Hurst and Sherman, combined
with a double expansion: a lace expansion for the lace-expansion coefficients.
