Yup, as soon as you receive solar power, the smart inserter will remove fuel from the boiler, so it will cut out sooner rather than later. (Hurrah for saving fuel!)
I didn't pick up from your screenshots that you use pumps like I have haha.
It would be neat to have relays connected to power ...
I find that the method you use above is very useful early game when you don't have large scale accumulator farms! I have a possible way of improving on your design, after realising you could control the operation of your steam engine banks through the use of electric pumps, as shown ...