The paper describes a way how one-sided Jacobi-type algorithm of Veselić for computing the hyperbolic singular value decomposition of rectangular matrices can be modified to work with blocks. The proposed modification preserves the relative accuracy property of the original algorithm and essentially improves its performance. Special attention is devoted to proving the global convergence of the method under some important classes of block-oriented pivot strategies. As numerical tests indicate, the block-oriented J-Jacobi methods combined with the Hermitian indefinite factorization, become efficient and accurate eigensolvers for Hermitian indefinite matrices.