We consider generalized interval exchange transformations (GIETs) of d intervals (d\geq 2) which are linearizable, i.e. differentiably conjugated to standard interval exchange maps (IETs) via a diffeomorphism h of [0, 1] and study the regularity of the conjugacy h. Using a renormalisation operator obtained accelerating Rauzy-Veech induction, we show that, under a full measure condition on the IET obtained by linearization, if the orbit of the GIET under renormalisation converges exponentially fast in a C^2 distance to the subspace of IETs, there exists an exponent 0 < \alpha < 1 such that h is C^{1+{\alpha}}. Combined with the results proved by the authors in [4], this implies in particular the following improvement of the rigidity result in genus two proved in previous work by the same authors (from C^1 to C^{1+{\alpha}} rigidity): for almost every irreducible IET T_0 with d = 4 or d = 5, for any GIET which is topologically conjugate to T_0 via a homeomorphism h and has vanishing boundary, the topological conjugacy h is actually a C^{1+{\alpha}} diffeomorphism, i.e. a diffeomorphism h with derivative Dh which is \alpha-Hölder continuous.
