Your Otis kit is just fine. Anything else possibly needed would be good solvent/cleaner and a good lube of your choice.
All a smaller BASIC shotgun cleaning kit needs is a small jar of powder solvent, half of an old cotton tee shirt, a bore snake, a small can/bottle of good gun oil (preferrably aerosol) and a phillips & regular screwdriver...and this all fits into a gallon zip-loc freezer bag. This will take care of most on the fly cleaning needs one may encounter while hunting. My large kit has other equipment for a complete breakdown, clean, lube & reassemble for after the hunt and prior to the gun going back into the safe.
For a totally complete kit get yourself a tackle box with the adjustable lure trays and a bottom section deep/big enough for cans of lube & solvents and keep everything you'll need in there. Mine has bore snakes in every gauge, brush rods & brushes of every caliber & gauge I own (I have a shotgun gun in every gauge from .410 through 12 and use a 10ga brush for chamber cleaning, plus rifles from 17 to 30 caliber and pistols from .22 to .45), pliars, screwdrivers, allen wrenches (standard and torkes style), loc-tite (for scope base mounting), Tetra Gun Grease, Choke Tube Lube, gun oil, Hoppes powder and copper solvent, foaming bore cleaner, a bore light, choke tube wrenches, an old tee or under shirt for rags/patches, a half dozen blue mechanics latex gloves and other misc items I use for maintenance, repair and cleaning of all my guns. I grab it when ever I go on a hunting weekend and can pack the smaller bag for a day on the water.
"Shoot low boys...they're riding shetland ponies in search of true grit" Lewis Grizzard