Careen Winters, executive vice president and global corporate reputation practice leader at MWW public relations firm, said that while it isn't a big deal to have a glitch in its registers, in the context of other issues that the retailer has underdone over the past few months, Sunday's event is further evidence there may be technology issues at Target.
"They need to be perfect right now," she said. "Target is under a microscope, and even a minor thing becomes a major thing."
Sozzi said the amount of customer outcry the glitch caused signals that Target's U.S. same-store sales—which fell 0.3 percent in the most recent quarter—continue to be affected by the breach. He added that they could continue to deteriorate if the company "doesn't clean its operationally sloppy self up." In its most recent earnings call, Target said it expects U.S. comparable-store sales to come in flat to up 2 percent for the year.
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But that's not all Target needs to do. Trout said the retailer needs to admit that, "'God damn, we've been having some trouble here,'" and outline that it has a very aggressive plan to fix its issues.
Target already has been taking steps to reassure its customers in the wake of the data breach. Last week, the retailer announced that it created a new role, chief information security officer. Earlier this year, Target appointed a new chief information officer and said it would roll out the new chip-and-PIN technology in early 2015, ahead of the October deadline when both banks and retailers need to be compliant with the technology. Experts say the technology is safer than the credit cards typically used in the U.S. because the chip prevents counterfeiting, while the PIN prevents fraud for lost or stolen cards.
CEO Gregg Steinhafel was also ousted from the retailer last month following the breach and a botched entry into the Canadian market.
But Trout said Target needs to do more than simply issue press releases about these changes. Instead, the retailer needs to run ads admitting that it lost its way, but has taken major steps to improve.
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J.C. Penney last year took a similar approach when it ran commercials apologizing to shoppers for the changes instituted by former CEO Ron Johnson, who steered the department store away from its discounting heritage and introduced designer labels that didn't resonate with its core shoppers.
"You've got to package it up a little better and make it more interesting," Trout said. "What you have to do is reintroduce what you would call the new Target."
This is especially true in the era of social media, Sozzi said. He said that platforms such as Twitter allow shoppers to become investigative reporters, and retailers aren't investing enough to manage their messaging on social media.
Winters sounded a similar note, saying that Target needs to stop being in a reactive mode. She said the company should talk, in particular, about the changes it is making on the technology front. The brand also needs to "double down" on its customer experience, including how the stores look, and continue to maintain its "cheeky, fun, fashionable" brand position, including partnerships with designers.
Trout did, however, give Target kudos for its attempts to quell angry shoppers. Although these efforts varied by location, some stores issued coupons, while others handed out free Starbucks drinks.
"If you can do something to make the line happier and less surly, absolutely [do it]," Trout said. "But it's better off to not have the lines."
—By CNBC's Krystina Gustafson, with contributions from CNBC's Courtney Reagan.