Dangers and threats, increased pressure and ticking clocks, loss of control, power struggles

There are a number of scenarios your character could face that will require them to stay hidden from a threat. The following conflict options will provide ample opportunities for high tension and close calls when they need to fly under the radar.

Being hunted by a predator (animal or human)
Having to sneak through enemy territory during a war or occupation
Hiding from an intruder during a home invasion
Being targeted for capture or execution (because the character possesses magic, has certain physical characteristics, belongs to a particular race, is divergent, etc.)
Avoiding someone who has targeted the character because their role in a prophecy poses a threat
Having to enter the witness protection program to hide from formidable enemies
Evading security during a break-in, break-out, or other prohibited activity
Meeting up with a contact at an event that's being surveilled by an enemy's security personnel 
Avoiding police (because the character has gone AWOL, is wanted for questioning, etc.)
Sneaking in (or out) after curfew
Needing to gain access to a lover's hotel room or apartment undetected

Being stuck in an uncomfortable position or a small space
Having to be very quiet and still
Having to wait (for a shift change, a mark to fall asleep, etc.) before leaving a hiding place
Being separated from loved ones
Having no one else to depend on
Being a naturally clumsy or loud person
Having to wear an uncomfortable disguise
Having to keep a supply of unusual resources to maintain the disguise 
Always having to be on the lookout for the enemy; not being able to relax
A loss of freedom (being confined to a room, a house, or specific location) until it's safe to move 
Nearly getting caught and having to lie, manipulate, or use threats to escape capture