It is known that a real-valued function defined on the unit interval is first-return recoverable if and only if itbelongs to Baire class one. Further, it is known that if first-return recoverability is replaced by stronger notions, such as universal or consistent first-return recoverability, then familiar subclasses of the Baire one functions are obtained. Likewise, if first-return recoverability is weakened to first-return recoverability except on a set of measure zero [first category], then one obtains precisely the class of Lebesgue measurable functions [functions having the Baire property]. Here we examine the situation where even smaller exceptional sets (countable or scattered) are excluded, and then explore possibility of combining these various methods for strengthening and weakening recoverability.