Okay, this doesn't really meet your requirements. Not much is absorbed through the skin. But mercury in your hand is fun to play with and your victim might keep it on longer than 4-5 seconds. The danger is from the vapors. If your plot allows, pouring large amounts of mercury around the victim's bed or car, or another relatively small space where the victim spends a lot of time, might do the trick.
Elemental mercury is what's in thermometers (the kind without batteries), so it would be relatively simple (though expensive and time consuming) to gather enough for the perpetrator to use (assuming a lot of safety equipment, the kind easily found online). The more dangerous form of mercury is methyl mercury, but that's not exactly sold in handy dandy glass containers. Mercury also has medical applications but purchasing it in larger quantities might prove more difficult.
Instead of buying old fashioned thermometers, the perp might want to buy new digital ones and offer them free to doctor's clinics and households and schools in exchange for mercury ones (as a way to reduce potential toxic exposures...swap programs like this probably already exist legitimately). You may also have some luck with fluorescent light bulbs.
Elemental mercury, also called “quicksilver,” is a heavy, silvery,
form of the metal mercury that is liquid at room temperature. It can
slowly change from a liquid into a gas that is invisible to the naked
eye. The gas or “vapors” that are released will start to fill a room
if mercury is spilled indoors.
Mercury is a very
toxic or poisonous substance that people can be exposed to in several
ways. If it is swallowed, like from a broken thermometer, it mostly
passes through your body and very little is absorbed. If you touch it,
a small amount may pass through your skin, but not usually enough to
harm you. Mercury is most harmful when you breathe in the vapors that
are released when a container is open or a spill occurs. (ref)
Will elemental mercury fumes kill a victim? Maybe. But organ damage (which you allow as an alternative) is more likely.
How much mercury spilled in a room will make air in the room unsafe?
Any amount of mercury spilled indoors can be hazardous. The more
mercury is spilled, the more its vapor will build up in air and the
more hazardous it will be. Even a small spill, such as from a broken
thermometer, can produce hazardous amounts of vapor if a room is small
enough, warm enough and people spend a good deal of time there, as in
a small bedroom. (ref)
Spilling the mercury on to a hot surface (like a stove) will vaporize it quickly. Spilling it elsewhere will allow the vapors to accumulate more slowly. The liquid form is easily tracked through a house and hard to get rid of.
Inhalation of elemental mercury vapors is the main source of toxicity,
as mercury is well-absorbed through the lungs. Problems from
inhalation result either from a large one-time high exposure or a
long-term exposure. Long-term exposure of inhaled vapors is generally
more dangerous, with the nervous system being the primary target of
mercury toxicity. Symptoms may occur within weeks but usually develop
insidiously over a period of years. Neurologic symptoms include
tremors, headaches, short-term memory loss, incoordination, weakness,
loss of appetite, altered sense of taste and smell, numbness and
tingling in the hands and feet, insomnia, and excessive sweating.
Psychiatric effects are also seen after long-term exposure. The
kidneys can also be effected. Intense exposure to high concentrations
of mercury vapor can lead to severe respiratory damage. (ref)
So there you have it. Organ damage from even short-term exposures. But not from being held in the hand. Consider that a frame challenge, that the substance you are imagining probably does not exist. But you can combine mercury in the hand with mercury spilling all over the place and creating fumes. Maybe try when the victim is locked into a sauna for an hour. There are many possibilities.