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We propose a new off-shell formulation for the massless N = 1 supersymmetric multiplet of integer superspin s in four dimensions, where s = 2, 3,.. (the s = 1 case corresponds to the gravitino multiplet). Its gauge freedom matches that of the superconformal superspin-s multiplet described in arXiv:1701.00682. The gauge-invariant action involves two compensating multiplets in addition to the superconformal superspin-s multiplet. Upon imposing a partial gauge fixing, this action reduces to the one describing the so-called longitudinal formulation for the massless superspin-s multiplet. Our new model is shown to possess a dual realisation obtained by applying a superfield Legendre transformation. We present a non-conformal higher spin supercurrent multiplet associated with the new integer superspin theory. This fermionic supercurrent is shown to occur in the Fayet-Sohnius model for a massive N = 2 hypermultiplet. We also give a new off-shell realisation for the massless gravitino multiplet.