You need to start somewhere and the initial point after mode-finding is usually preferable to other points.
Theory tells us that the Metropolis-Hastings algorithm will usually work with any positive definite proposal density covariance matrix, however using an poor one will decrease efficiency. Using the Hessian at the mode is typically very efficient, but as you saw, it is sometimes hard to obtain. In that case, you can tell Dynare to use a different matrix using the mcmc_jumping_covariance-option. For example, you can use mcmc_jumping_covariance=identity_matrix
That being said, only do this if you are sure that the invalid Hessian is not due to deeper problems in the model like non-identified parameters or wrong observation equations.