Geometric descriptions for locally complete families of projective hyperkähler manifolds are only known in very few cases. In this talk, we first describe the projective geometry of the Hilbert square of a K3 surface of genus 7 or 8, by making use of the Mukai model: in both cases, it can be realized as a degeneracy locus on an ambient homogeneous space. From this, we deduce a geometric description for the two locally complete families of K3^[2]-type (square 4 and square 6 with divisibility 1), in terms of Coble type hypersurfaces. If time permits, I will present the construction of an explicit one-dimensional family admitting a large finite automorphism group, as a baby example towards understanding the whole moduli space. Based on a joint work with Ángel Ríos Ortiz and Andrés Rojas, and an ongoing work also with Benedetta Piroddi.