We consider the solution of large linear systems with multiple right-hand sides using a block GMRES approach. We introduce a new algorithm that effectively handles the situation of almost rank deficient block generated by the block Arnoldi procedure and that enables the recycling of spectral information at restart. The first feature is inherited from an algorithm introduced by Robbe and Sadkane Linear Algebra Appl. , 419 (2006), pp. 265--285, while the second one is obtained by extending the deflated restarting strategy proposed by Morgan Appl. Numer. Math. , 54 (2005), pp. 222--236. Through numerical experiments, we show that the new algorithm combines efficiently the attractive numerical features of its two parents and outperforms them.