I think we're now at the self-sustaining phase of the heatwave. It's a bit like in winter, where you need low 850s and deep cold air to start a cold spell, but once you've snow on the ground the boundary layer chills enough that you can get away with relatively higher 850s to sustain it.
The 12z GFS shows this in its summer guise. The 850s aren't anything to write home about, but there are areas of 30 or more every day from tomorrow out to day 9 (as far as the high-res charts currently go). The boundary layer heats enough to generate small areas of 14 or 15C 850s during the late afternoon over the warmest areas, but 850s are generally lower aside from that. As a rule of thumb, at this time of year under clear skies and low winds, and for inland locations, you would add 16C to the 850s to get a rough estimate of surface temperatures, so 14C for 30, 24C for 40 and so on.
The dryness of the land is doubtless involved with this, as dry land heats up that much faster.