The problem of drift estimation for a process X solution of stochastic differential equation with Lévy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the parameter of the drift is constructed under conditions relating the jump behavior and the sampling scheme. n∆^{3−ε }_n → 0, where n is a number of observations and ∆n is the maximal sampling step. This condition relaxes the condition n∆^2_ n → 0 usually required for joint estimation of drift and diffusion coefficients for SDE with jumps. The main challenge in this estimation problem stems from the appearance of the unobserved continuous part X^c in the likelihood function. In order to construct the drift estimator, we recover this continuous part from discrete observations. More precisely, we estimate, in a non parametric way, stochastic integrals with respect to X^c. Convergence results, of independent interest, are proved for these nonparametric estimators. Finally, we illustrate the behaviour of our drift estimator for a number of popular in finance Lévy–driven SDE models.