We study the number of simultaneous diagonal flips required to transform any triangulation into any other. The best known algorithm requires no more than 4×(2log5453+2log65)logn+2≈327.1logn simultaneous flips. This bound is asymptotically tight. In this paper, exploiting some newly discovered properties of blocking and blocked chords we bring the leading constant down to ≈85.8. We also show that 6lognlog97+1≈16.6logn simultaneous flips are sufficient to transform any maximal outerplane graph into any other.