Every subinterval provides in addition the $2n$ conditions $$y_i = s(x_i),$$ and $$s(x_{i+1})= y_{i+1},$$ to be fulfilled. If we also assume that $s'$ and $s''$ are continuous, then $$s'_{i-1}(x_i)= s'_i(x_i),$$ yields $n-1$ conditions. Similarly, $$s''_{i-1}(x_i)= s''_i(x_i),$$ results in additional $n-1$ conditions. In total we have $4n$ coefficients and $4n-2$ equations to determine them, leaving us with $2$ degrees of freedom to be determined.