Long story short, in The Door Into Summer, the main character freezes himself and wakes up in the future. Then he goes back to the past using an experimental time travel, does his clever business, and comes to the same cryo facility to sleep back to the future. Wouldn't the staff of the cryo company be surprised when 2 Dan Davises show up on the day of contract to be freezed?
He doesn't use the same facility the second time. The first time, Belle changes his paperwork for Mutual Assurance to instead go to Master Insurance (same number of characters since they had to do this on a typewriter, and a subsidiary of Mannix, which Belle used to work for and therefore had contacts) and has him checked in there. The second time, he goes to the facility that he was originally going to use the first time, Mutual Assurance.