Do you have access to above the ceiling?

I don't think using that type of duct is best for range hoods because the grooves can catch the grease, and also the air flow is not as good as smooth solid duct work. Our previous venting was even worse than that because it was flexible (like for a dryer vent), so we replaced everything. But we had access to everything in our attic.

There are some videos on youtube that have instructions for installing hoods and some of them might help you. But you should make sure what you are doing is safe and the installation is the best it can be so your hood is effective.

This one shows at least how to connect the duct pieces to each other. https://www.youtube.com/watch?v=9egwOazPJis

Hi Lisa, thank you for the advice! There is no attic access from up the ceiling, so if I want to replace the whole thing, the ceiling needs to be opened up from beneath. :( I saw the video you suggested as well from this old house, I guess it's just a simple taping process. Didn't think it would be that simple. :D

You can see the remnants of mastic on the old duct. It's not the correct material. You need to use metallic tape over the joint between the new and old. One should slide into the other. Some people will join the two pipes with a sheet metal screw, also.

Contrary to it's name, duct tape is not appropriate to use joints. It degrades too quickly and won't handle the heat.