For emphasizing the action of tidying:
* Re-: (e.g., re-tidy - to tidy again)
* Un-: (e.g., un-tidy - to make something messy)
* Pre-: (e.g., pre-tidy - to tidy beforehand)
* Over-: (e.g., over-tidy - to tidy excessively)
For describing a state of being tidy:
* Neat-: (e.g., neat-tidy - very tidy)
* Spick-and-span-: (e.g., spick-and-span-tidy - perfectly tidy)
* Immaculate-: (e.g., immaculate-tidy - spotless and perfectly tidy)
* Well-: (e.g., well-tidy - neatly tidy)
For emphasizing a specific type of tidying:
* Spot-: (e.g., spot-tidy - to tidy a specific area)
* Quick-: (e.g., quick-tidy - to tidy quickly)
* Deep-: (e.g., deep-tidy - a thorough cleaning and tidying)
Ultimately, the best prefix depends on what you're trying to convey.