Nearly everyone says let me know if there's anything I can do, and nearly nobody gets taken up on it, because the offer is shaped in a way that can't be accepted. Anything is too big to ask for.
What works is a specific instruction. Can you pick up a prescription on Thursday. Can you sit in the room while I make this call. People are enormously relieved to be handed a task; it's the vagueness that leaves them stranded on the doorstep.
Asking small isn't imposing. It's the mechanism by which offers already made turn into something real.