We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2$s$, with $${1\leqslant s\leqslant k^{2}-1}$$ . In this way, we prove the main conjecture for such even moments when $${1\leqslant s\leqslant \tfrac{1}{4}(k+1)^{2}}$$ , showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of$s$, these finding application to the asymptotic formula in Waring’s problem.