It depends on what you want to do. A lot of the long-distance races have tandem categories. These are typically 6-hour, 12-hour, 24-hour or 500-mile type distances. In a lot of them, you will be the only tandem, so if you're looking for actual competition, that's something different. Here's one, for example, 12 hour and 6 hour with tandem categories in both: HOME - Maryland Endurance Challenge Look up others at www.ultracycling.com


The Texas Time Trials in Glen Rose, Texas (southwest of Dallas) is a 500 mile/24hr/12hr/6hr event that includes tandems.
The Texas Time Trials Championship is a 40k time trial in south Texas that also includes tandems. I've done the first, know some people that have done the latter- it had three tandems in it that year.