No, you don’t need additional carbon taxes. Farmer Bob bought his tractor diesel from Exxon, which already bought the carbon credits when they refined the crude into fuel. The carbon cost is already built into the price.
The credits should be required as far up the “food chain” as possible as long as the result is likely to be emitted eventually. Any type of fuel is obviously going to be burned at some point.
You’ll have an equal amount of whining with a tax. Certain industries will ask for an exception, or say it’s too high, or whatever. This is inevitable.
Cap-and-trade however ensures that the carbon goes to those industries that need it most, thereby minimizing the economic disruption.