Well, you’re going to have to settle on what kind of stick you’re using: such as a baton or staff because length is important. You’re also going to have to designate what kind of staff fighting they’ve been trained in because techniques differ based on style. Depending on the era, a secret agent may get more use out of Filipino Eskrima stick techniques than he or she would out of a bo staff because one can be hidden fairly easily while the other is more common. A spy does not carry weapons that make them stand out in a crowd. A staff is a long arm and thus a very tall weapon, if the spy does not exist in a culture where carrying a staff is common (such as modern day US) then they will forgo it entirely.
A spy may carry a staff in a historical setting when they were common, such as Medieval Europe, or appropriate corresponding periods in China and Japan when looking to pass for a specific kind of peasant or soldier.
However, depending on where they are going, a spy may choose to carry a tactical baton on their person. A tactical baton is a police weapon, it folds up easily and can slip into a pocket. However, they are illegal to own in most states without a permit. It’s also meant for tagging someone in the back of the skull. It is meant to hand out a single pounding, not take a pounding from another weapon.
Staffs and sticks are bashing weapons. They don’t stab or cut, instead they cause blunt force trauma. They are very effective weapons. However, if you’re feeling cheesy, a wooden stick can be snuck past a metal detector (but not an x-ray machine) and people whether it’s onlookers or guards will remember the person who fought with the wooden stick because it’s unusual.
If you want to have a secret agent who specializes in stick fighting in the modern day, then they only have one real option: a cane. An umbrella would also work, but many people will wonder why your spy is carrying an umbrella if they are traveling to a place where rain is not a constant. A wooden cane is justifiable, easy to explain, and carry into places that normally would not allow them. It can also be taken places where a tactical baton would be discovered and removed.
However, while a cane is an excellent self-defense weapon, it can get tricky in modern combat if you’re spy is not acting via surprise. A cane does not have an advantage against a gun that is already drawn. A cane has a reach advantage against someone with a knife, but lacks being comparably lethal. The kills will also be fairly recognizable and they will take longer.
The short of it is: while training in combat with sticks is justifiable for a spy, it’s unlikely to be their specialty. In the modern world, a spy needs to be trained on a variety of guns, knives, and in a variety of hand to hand styles to be successful. A cane may look odd on someone who does not fake a limp or is not elderly and a fake limp is difficult to maintain long term. If a character is using the cane to support their own weight because their leg can’t maintain it on it’s own then they’ll have difficulty fighting with it. (This won’t stop someone with a limp from being a spy.) In the modern world, it’s not a sharp choice because there are better ones available.
Remember: a spy isn’t looking to stand out, they want to blend in and disappear. Pick styles and weapons that fit into the frame of who they are pretending to be and ones that reflect the culture (and social status) they exist in.