Joe Corbett, the USPS's chief financial officer, said it would be "very short-sighted to think our problems are solved at this time."
Without the legislation, officials are insistent that the agency will continue to experience multi-billions in annual losses.
Donahoe reiterated that the USPS's five-year plan does provide at least $20 billion in savings and that he expects some stability from the plan, which also is contingent on continued growth of the economy and improvement in the labor market.
Corbett said the Postal Service's cash position would become severely hampered in mid-October when it is required to make its annual payment of approximately $1.4 billion to the Department of Labor for workers’ compensation.
That will leave the agency with about five days of operating cash on hand, which Corbett called "dangerously low" and should be a harsh reminder that by no means is the Postal Service in the clear.
Donahoe said that five days of cash "is as much of a crisis that we've been in."
On Friday, the USPS reported its losses of $3.9 billion so far this year, putting the agency on track to come in well below last year's record losses of $15.2 billion.
The agency is still expected to record about $6 billion in losses this year, although postal officials said it is difficult to gauge because of the volatility of interest rates on the worker compensation adjustment.
The USPS had a net loss of $740 million in the third quarter, which ended June 30.
Officials insisted everyone has to "get their head out of the sand" and understand the digital world's effects on the bottom line.
Still, most of the losses stem from a requirement to prepay the healthcare of future retirees.
"We hope the public and the mailing industry will read between the lines and understand that this quarterly report is encouraging news, including the sharp rise in package delivery revenue, despite efforts to perpetuate a doomsday scenario and push Congress to reduce services," said Fredric Rolando, president of the National Association of Letter Carriers
"These results make it clear that lawmakers need to concentrate on fixing what’s broken with the Postal Service – including the pre-funding mandate and the governance structure – while strengthening what’s working: The world’s most affordable delivery network."
A bright spot in last week's report showed that package revenue was up 8.8 percent over the third quarter of 2012, along with in increase in online shopping.
Meanwhile, first-class mail revenue declined 0.9 percent because of a 3.4-percent decrease in mail volume.
Sen. Bernie SandersBernie SandersFive takeaways from Pruitt's EPA hearing Overnight Energy: Trump's EPA pick faces Congress | 2016 is hottest year on record Five takeaways from Price's confirmation hearing MORE (I-Vt.) argued on the Monday that the Postal Service would have posted a profit for the past three months without the requirement to deposit $5.5 billion a year into the retiree fund.
“The good news is that the Postal Service made a profit except for the unfair, onerous burden that no other company or agency is required to pay,” Sanders said.
"The Postal Service has experienced financial problems and changes need to be made to bring it into the 21st century, but I am convinced that proposals to slow down mail delivery and provide less service are the wrong way to go."
Sanders introduced legislation in February that would ensure Saturday mail continues and repeal the law that requires the massive payments into the retirement fund.
House and Senate legislation in the works does aim to change the prepayment requirement and give the agency greater flexibility to cut costs while boosting its revenue.
But lawmakers have yet to iron out their remaining differences after two years of attempts to send a bill to President Obama's desk.