His 'secretary' wasn't there in the beginning
She was gone for lunch
Bit of a giveaway there
A giveaway? How? It's not unusual for secretaries to have lunch outside the office; in fact, it's perfectly natural. And since Reggie arrived for her appointment with Mr. Bartholomew at half past noon - an appropriate time for lunch - there's nothing suspicious about the secretary ("Miss Tompkins") not being there.
Actually, I thought the ruse cooked up by Walter Matthau's character was pretty clever; he found an office at the embassy he knew would be empty & even used official Foreign Service stationery to contact Reggie.