Rodent poison is slow acting (so they take it back to nest & poison their young). The problem is that they sometimes crawl away & die in inaccessible places - but the foul smell usually disappears after a week or so once it's decomposed - quite quickly once flies find it and the maggots do their thing. Sometimes takes a bit longer and the smell lingers if it's cold, quicker but more intense in hot weather or if it's crawled under the floor and died on the heating pipes. When our pest-control contract comes up for renewal, I'm going to seriously look into rat-traps rather than bait - though it's more labour intensive & costly as means someone has to go check the traps several times a week. With the poison, I just go on the hunt when my nose tells me to.
I'm not sure that rats leave trails - mice do, they dribble urine as they scurry about. Rats are smarter and may have just come in for the charcoal, or you may have some other food source in there that you've forgotten about.
If you do use a trap, a pest controller told me once that best thing he'd found to use were Rolos - they're attracted to the chocolate and toffee and it's sticky enough to keep them from doing a quick grab that doesn't set the trap off.
Sorry if this it too much graphic detail - I've had more experience than I care for, of both mice and rats.
