I live in Arizona and have ridden many of the following areas. There are reasonable alternatives to Route 288 to get to the Mogollon Rim from Silver City. One would be to take US-70 to Globe, AZ-88 to Roosevelt, AZ-188 to AZ-87 to Payson. From Payson, you could ride AZ-87 to the Rim Road. From that point, you could go west through Camp Verde to Prescott (which involves a LOT of climbing), then north to Ash Fork and ride I-40 or much of old Route 66 to the California border. I'd recommend riding from Strawberry to the Lake Mary Road which goes north to Flagstaff. From Flagstaff, you can ride most of the way to the CA border on old Route 66. Several websites identify parts of Route 66 that exist but are not shown on most maps.
An alternative route to the Rim Road through scenic backcountry is on US-180 from Silver City to Eagar, AZ, then AZ-260 though Show Low to the Rim Road. There are few services, other that some at Glenwood, between Silver City and Alpine.