We propose a new spline collocation-based approach for solving Volterra integral equations of the first and second kinds. The solution is represented as a linear combination of quadratic minimal splines of maximal smoothness, with the coefficients determined through specialized local approximation schemes (quasi-interpolation). Numerical experiments demonstrate that the use of non-polynomial splines leads to higher accuracy of the approximate solution compared to previously proposed approaches based on -splines, including approaches defined on nonuniform grids.