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In the present paper, we numerically construct new non-topological, spontaneously scalarized neutron stars in the tensor-multi-scalar theories of gravity whose target space is a three-dimensional maximally symmetric space, namely either $\mathbb{S}^3$, $\mathbb{H}^3$ or $\mathbb{R}^3$, and in the case of a nontrivial map $φ: \text{\it spacetime} \to \text{\it target space}$. The theories of gravity admitting scalarization are characterized by the fact that the field equations always admit the general relativistic solution but for certain ranges of the parameters space it loses stability and nonlinear development of a scalar field is observed. Thus, in order to determine the values of the parameters where such scalarization is possible we studied the stability of the general relativistic solution within the framework of the considered tensor-multi-scalar theories. Based on these results we could obtain a family of scalarized branches characterized by the number of the scalar field nodes. These branches bifurcate from the general relativistic solution at the points where new unstable modes appear and they are energetically more favorable over the pure Einstein solutions. Interestingly, in certain parameter ranges, we could obtain non-uniqueness within a single branch of scalarized solutions.