The mystery sender? The florist itself, Bloom & Wild Ltd.—or, more specifically, the florist’s “customer delight” department, a team hired to put customers’ floral grievances right and surprise deserving members of the public with the gift of the garland.

“We give every member of our customer delight team a budget to look for opportunities to proactively go out of the way to do something nice,” said Aron Gelbard, Bloom & Wild’s co-founder and chief executive officer. The company, which declined to give details on the size of the customer delight budgets, employs 22 people on the team, staffing up with seasonal workers at peak times such as Valentine’s Day and Mother’s Day.

The companies that hire them say customer delight associates are not just support staff with a quirky name. Instead, they aim to provide a sort of customer service on steroids—swiftly and personally responding to inbound complaints and queries, as well as looking for ways to surprise consumers with something like a personal note or a free product.

The role is still relatively uncommon. LinkedIn said not enough of its users have the job title to statistically analyze how it has grown in popularity. But the idea of delighting a customer, rather than simply serving them, is not new in service industries such as hospitality. Employees at the best-rated hotels for years have been taught to learn guests’ names, and smile when interacting with them, to provide an experience that is predictable and consistently above average, said Shep Hyken, a customer experience consultant.

The form of customer delight adopted by e-commerce startups like Bloom & Wild, meanwhile, aims to set companies apart by improving the often nameless, faceless and automated work of online customer support. Those who take on the job enjoy more autonomy and variety, and better pay, than their peers in standard contact-center roles, their employers say.

Not all customers are interested in being delighted by e-commerce businesses in the same way they might expect to be delighted by a swanky hotel—most just want things to work, and the ability to speak to a knowledgeable human when they don’t, Mr. Hyken said. He added that online shoppers really want convenience—not delight.

Besides, companies—particularly large or growing ones—“cannot be above and beyond the whole time. It’s just not sustainable,” Mr. Hyken said. And spending money to elevate customer service to delightful service can be at odds with the growth of e-commerce businesses.

“There have been times we’ve intentionally had to scale back, knowing that the relationship with the customer comes first,” said Lara Casey Isaacson, CEO of Cultivate What Matters, a stationery company that has six customer delight employees. The online retailer held back expanding into new markets, for instance, because Ms. Isaacson did not want to partially automate or outsource customer care—options that investors told her would make expansion more viable, she said.

Cultivate’s customer delight associates are trained to use positive language in all communications (it’s “my pleasure” instead of “no problem,” and always “and” instead of “but”), and, as at Bloom & Wild, they are given a budget. They have sent coffee-shop gift cards to new students buying planners and flowers to customers who have written to say they are using Cultivate’s planning products to get them through a tough time, Ms. Isaacson said.

Customer delight can also take the form of damage control. The roughly 100 customer delight associates at Zomato, a restaurant recommendation and delivery service based in India, are authorized to purchase replacement meals and gift boxes of chocolates and cupcakes for customers whose orders got lost or damaged in transit, in addition to a refund, said Surobhi Das, Zomato’s head of customer experience. One of their main jobs, however, is to pre-empt any disruption to delivery, and get on top of it before the customer thinks to pick up the phone to complain, she said.

“If I can see that a customer’s order is getting delayed, can I proactively call up the restaurant, or the rider, and ensure that the food reaches as close to the delivery time we promised as possible?” Ms. Das said. “There is nothing more delightful than the customer not even having to reach out to support to complain about an issue.”

Blendily, a botanical skin-care business based in Portland, Ore., in July hired its first customer happiness associate. Part of the job involves placing “happiness calls” to customers 30 days after their purchase.

“It’s just an old-school, pick up the phone and call to see if they like it routine, and if they don’t, then we suggest something else,” said Blendily founder Ivy Chuang.

Some customers are confused when they get follow-up calls, Ms. Chuang said. In fact, some are bemused to hear about companies’ customer delight departments altogether—especially when they are anything but delighted by the service provided.

“If you’ve got a team whose job it is to deal with complaints from angry customers, calling it ‘customer delight’ is only destined to make those people even more cross,” said Rhodri Marsden, a London-based writer and musician who received an influx of marketing emails from Bloom & Wild’s customer delight team after sending flowers to a friend.

Mr. Gelbard of Bloom & Wild said he is aware that the quaint term “delight” may be ripe for lampooning online. But that isn’t necessarily a bad thing, he said.

“Setting our standard as delight rather than service encourages more complaints, which means we have more data, which means we can start to spot patterns,” Mr. Gelbard said. “And the more rapidly we can spot root causes, the more rapidly we’ll be able to address them.”

Write to Katie Deighton at katie.deighton@wsj.com