[Big Dave]

What you describe is called compounding.

Compounding can indeed increase eficiency. The old B-29 engine (R-3360 I think) and the Napier Sabre were some of the most efficient engines ever built.

Compounding works well (with some reliability issues) at high engine load, but has no scope for contribution at the low loads used by us MPG seekers. Consider turbo lag. A tirbocharger is a form of compounding where the waste heat engine (the turbo) drives a centrifugal compressor. At a low gas flow and heat rate, the turbo makes almost no boost. At high gasflow rate(high RPM) and high heat rate, it works well enough to require a wastegate to prevent overboost.

For a car, with its very light duty cycle to be able to use compounding the main engine would have to be so underpowered as to be running at high load a high percentage of the time. It ain't worth the effort on a high-MPG car.