You're probably right. I think I have good intake ventilation though between my two gable vents and my solar attic fan. It wouldn't hurt to test it out with the attic hatch shut tight. Do you have much confidence in those hatch covers that are designed to fit over them?
Most of those covers are insulating only, not sealing. Your present hatch cover should have drywall underside (or some other fireproof/fire retardant material), have some kind of seal or weatherstripping around the outer edge to match up to the hatch opening, and be heavy enough so the weatherstripping has a good seal. Then you can use an insulating hatch cover over that.
I have attic stairs that have insulation built in, for about R6. That's enough for my climate, but if you live in a cold climate you might want an insulating hatch cover. You can build your own out of rigid foam insulation they sell at big box stores and lumber yards.