The same paper shows that the converse is true: starting from time-ordered Green functions satisfying axioms T1-T7 as in subsection I.1 one can get the Schwinger functions (Theorem 1, pp. 99, add Corollary 3 from same page to get full Euclidean covariance S5 of Schwinger functions from Poincaré covariance T8 of time-ordered Green functions). Since these will then satisfy a strengthened form S1-S4 (+S5 if T8 holds) of the Osterwalder-Schrader axioms (see subsection I.2) to allow for coinciding points, the Osterwalder-Schrader reconstruction theorem applies and from them one can recover the Wightman field operator-valued distributions and the corresponding Wightman functions.
Edit: I believe an explanation is in order as of why the roundabout path through Schwinger functions, albeit natural in the context of the paper by Eckmann and Epstein cited by the OP, is in a sense necessary. The strategy outlined in the previous paragraph has the time-ordered Green functions by themselves as its starting point. If one considers that these are the vacuum expectation values of time-ordered products of a Wightman field - or, put differently and perhaps more appropriately, the time-ordered (part of the) Wightman functions -, one is faced with the problem of how to define these from the Wightman axioms alone. The problem is, since the time-ordered Green functions are obtained from the Wightman functions by multiplying the latter with products of Heaviside step functions in time, the Wightman axioms are not strong enough by themselves to control their singular behavior in order to guarantee that such multiplications are well defined.
The problem posed in the previous paragraph was addressed by Steinmann (Zur definition der retardierte und zeitgeordneten Produkte, Helv. Phys. Acta 36 (1963) 90-112) and it is essentially a renormalization problem in the same sense as in formal perturbative quantum field theory, that is, it is an extension problem for $n$-point tempered distributions to the so-called large diagonal of $\mathbb{R}^d$ $$\widetilde{\Delta}_n(\mathbb{R}^d)=\{(x_1,\ldots,x_n)\in\mathbb{R}^{nd}\ |\ x_i=x_j\text{ for some }i\neq j\}\ ,$$ for time-ordered Green functions are a priori well defined only in $\mathbb{R}^{nd}\smallsetminus\widetilde{\Delta}_n(\mathbb{R}^d)$ due to their Lorentz invariance. As such, Steinmann only managed to solve it for two space-time dimensions (for the two-point functions, for all space-time dimensions). The higher dimensional case remains open to this day. Of course, one can apply the Hahn-Banach theorem to get an extension of the time-ordered Green functions to all of $\mathbb{R}^{nd}$, but one needs to find an extension (possibly as a modification of a generic extension provided by Hahn-Banach) retaining the additional properties one formally expects from time-ordered Green functions - Lorentz invariance among them. This is formalized e.g. by axioms T1-T8 of Eckmann-Epstein. For use in axiomatic quantum field scattering theory, Haag, Ruelle and Araki use regularized step functions instead to circumvent this problem, see e.g. Chapter 5 of the book by H. Araki, Mathematical Theory of Quantum Fields (Oxford University Press, 1999).
The extension problem to the large diagonal for time-ordered Green functions is, of course, the same one faces when defining Schwinger functions from the Wightman functions. As such, one is faced in both cases with the problem of non-uniqueness of such extensions, even when one requires e.g. the Eckmann-Epstein axiom sets T1-T8 and S1-S5 respectively. As a rule, in the case of the $n$-point Schwinger functions there is a distinguished choice of extension if they are the $n$-point moments of a Borel probability measure on $\mathscr{S}'(\mathbb{R}^d)$, for these moments are necessarily well defined on the whole of $\mathbb{R}^{nd}$ if they exist at all. This is usually the case in models, as discussed in Section IV of the paper by Eckmann and Epstein for the $\varphi^4_3$ model. For time-ordered Green functions, one expects the same from formal renormalized QFT perturbation theory but admittedly there is (yet) no clear, rigorous direct prescription because most of rigorous non-perturbative QFT model building is done in the Euclidean domain precisely by constructing such measures, so in this sense the Schwinger function detour taken in the first paragraph is unavoidable.
