It usually is easier to shovel out the old partially burnt coal and start from clean grates. Then once you have established a good hot coal fire, toss the 'old coal' on the fire,, it will have the best possible opportunity to burn on the top/center of the fresh fire.. Sometimes the unburnt remaining coal won't burn very well on the bottom of the fire, and these pieces of unburnt coal can be difficult to get ground up by the grates and into the ashpan.. if too much of this large, unburnable stuff accumulates, then getting enough air to the fire can become a problem. As well as jamming in the grate openings while shaking.
Once you decide to burn for several days or weeks, or for the season,, load your firebox to the top of the firebrick,, the coal will burn better in a deep coal bed, more completely, and it will be easier to control, and you can even slow the burn rate more with a deep fire than with a shallow fire..
If I were to build a hand-fed firebox, now that I've made several mistakes and struggled with poor designs,, I'd make the firebox at least as deep as it is square,, so it would be probably 16"x16"x 20" deep.. with all the bottom of the fire being shaker grates.. Firebrick lined. If I were to make it round, it would have the rough dimesions of a 5 gallon bucket..
Hope this helps.. Greg L