A new expression is found for the spin-dependent contribution Δms to the self-energy of an electron moving with a transverse momentum p⊥ in an electric field. The structure of an asymptotic expansion of Δms(r, χ) as a function of two dynamical invariants r = γ⊥−2 and χ = γ⊥|ɛ|/ɛc (γ⊥2 ≡ 1 + p⊥2/m2c2 and ɛc ≡ m2c3/|e|ℏ) is clarified with the aid of this expression. The expansion of Δms(r, χ) can be represented in the form of a Taylor series in r, the coefficients in this series, F0(χ), F1(χ), etc., being integrals of the Mellin type. The major coefficient F0(χ) is universal and, in the case of an appropriate interpretation of χ, describes well-known spin-dependent corrections to the mass in three different cases of a constant external field (for r → 0). The asymptotic properties of the function F1(χ) are studied in detail, but only order-of-magnitude estimates are obtained for F2(χ) and F3(χ). A comparison of these contributions revealed that, in the semiclassical region χ ≪ 1, r is indeed the parameter of the aforementioned expansion, while, for χ ≫ 1, the true parameter is rχ2 ≡ β2. In particular, the anomalous magnetic moment develops, owing to F1(χ), a term that grows logarithmically for χ ≫ 1, but which does not violate the hierarchy of terms in the Taylor series being considered, provided that β remains smaller than unity.