How tall can the hood be to fit under your cabinets? Vent-A-Hood makes a 9" tall under-cabinet hood in a number of styles that can be vented straight out the back. Unfortunately, 9" tall models are only 21" deep, so it won't extend out over your entire cooking surface.

If an 18" tall model would fit, then you have a choice if 24" or 27" deep models. The latter would be preferable to extend out over the entire cooking surface. The model PRX-18-242 is a 27" deep pro-style 600 CFM model that allows the option to be vented straight out the back. Higher CFM models are available, but 600 CFM should be adequate over your range.

I have a 48" 600 CFM Vent-A-Hood over my 36" cooktop. I've been very pleased with it. Mine vents straight up through the roof and is very effective and quite unobtrusive from a noise perspective. I have heard, however, that venting straight out the back is not quite as conducive to quiet operation. The fit and finish on Vent-A-Hood products is excellent with no sharp edges, making cleaning a simple matter.

Vent-A-Hood has several interesting videos on their website about how and why they think their extraction system is superior to other brands. You might find them interesting, then you can form your own opinion.

Here is a link that might be useful: Vent-A-Hood PR-18-242