The two-level local projection stabilization is considered as a one-level approach in which the enrichments on each element are piecewise polynomial functions. The dimension of the enrichment space can be significantly reduced without losing the convergence order. For example, using continuous piecewise polynomials of degree $r \geq 1$, only one function per cell is needed as enrichment instead of $r$ in the two-level approach. Moreover, in the constant coefficient case, we derive formulas for the user-chosen stabilization parameter which guarantee that the linear part of the solution becomes nodally exact.