When not wanting to drive themselves to a given destination, some passengers — be they faithful to Uber or Lyft — feel the need for speed.

But I feel the need for Ameed.

Ameed Musleh is an independent driver for hire. Although he either owns or has access to newer Lincoln Town Cars, his primary vehicle is a 1998 edition with nearly 400,000 miles on the odometer.

Its back seat and trunk are extraordinarily spacious, and it floats along the road as though you’re on a lake with no chop. Its black exterior is exquisitely maintained; its tan interior is cracking a bit and smells of cigarettes, but is still plenty comfy.

If you want to ride in silence, Mr. Musleh, 50, will grant you that serenity without you having to specify as much on an app. If you like your conversation a bit bawdy, Mr. Musleh can work blue. Standup comics read a room; Mr. Musleh reads his back seat.

Occasionally, Mr. Musleh is available on extremely short notice. But more often than not, you have to book him a few hours in advance, pre-negotiating a rate that almost always beats those of the ride-share services.

On the back end of a big night on Seattle’s bustling Ballard Avenue, while scores of tipplers wage war with their smartphones in hopes of landing a limited supply of Ubers or Lyfts in the immediate area, Mr. Musleh’s Town Car pulls smoothly up to where you’re standing on the side of the road. It looks particularly regal in the rain, and it only has eyes for you.

Even with the predominance of Uber and Lyft, plenty of Americans still ask to be driven someplace by a “town car.” But with every passing year, the odds become slimmer that they’ll be ferried to their destination by an actual Town Car, a model Lincoln ceased manufacturing in 2011, and which is now too old to qualify for Uber and Lyft’s luxury offerings — or, in some major cities, their standard level of service.

And because of the deteriorating interiors, many private livery companies are slowly phasing out even newer Town Cars, replacing them with supersize S.U.V.s, Mercedes sedans, Chrysler 300s or Lincoln Continentals, a model that Lincoln rebooted in 2016 after a 14-year absence from new-car lots, presumably because Matthew McConaughey wanted to make weird commercials about them.

“A lot of people will, de facto, be like, ‘I need a Town Car,’” said Colin Perceful, a former doorman at the Muse Hotel in New York and the Four Seasons in Seattle, who now operates his own tour company, Totally Seattle Tours. “We’ll say, ‘O.K., how about a luxury sedan?’ And they’re like, ‘Yeah, same thing.’ It’s like when someone asks for a Kleenex instead of a tissue. The Town Car sort of embedded itself into the lexicon of transportation.”

A native of Pakistan, Mr. Musleh moved from a refugee camp in Jordan to the United States three decades ago on a student visa. He had a friend who drove a taxi, so he tried that for about 10 years.

But he felt as if he was being charged too much to lease his vehicle from a cab company, so as soon as he had enough regular customers, he plunked down $5,000 for that ’98 Town Car and struck out on his own after briefly considering driving 18-wheelers for a living.

Business boomed for the first five years, he said, before Uber really asserted itself and cost Mr. Musleh about 75 percent of his regular business. Hence, he is considering adding a newer Chrysler 300 to his fleet so he can augment his income by driving for Lyft.

“Fifty percent of my customers left because they got older and don’t go out as much, 25 percent went to Uber and Lyft, and 25 percent remained,” Mr. Musleh said. “Young customers feel like they’re bothering me.”

Curiously enough, when Uber got its start in 2009 as UberCab, it was primarily a black-car service that relied heavily on Lincoln Town Cars. But with the dawn of UberX, the cheaper service, the company’s hybrid-heavy fleet effectively ate its dad.

In a way, this is just history repeating itself. Around the time Geoff Puett got into the chauffeured car business in the mid-’90s, industry preferences were shifting from stretch limousines to Lincoln Town Cars, which, as he said, adopted a “smaller, rounder body style” in 1998.

Mr. Puett, the general manager of Bayview Limousine in suburban Seattle, still has five 2011 Town Cars in his fleet because, as he put it, he “has passengers who refuse to ride in anything else.”

By contrast, Bayview has 17 Continentals, a number that may increase once the company sells off its Town Cars. He likes the new Continentals but conceded that the trunk isn’t nearly as deep as the Town Car’s, joking that “you could get five bodies” in that storage space.

When Mr. Puett has had to sell stretch limos to private buyers, he has been struck by the number of families who have snapped them up. Turns out that the window between the front and back seat is not only useful for concealing morally questionable behavior among adults, but it’s also great for deadening the din of screaming 6-year-olds.

While Mr. Puett pointed out that later-model Town Cars were virtually “all fleet sales,” there are some passionate individual enthusiasts.

“They were and are bullet cars that can go 300,000 miles easily,” said David Gustafson, the 80-year-old communications chairman of the National Lincoln and Continental Owners Club. A resident of Burnsville, Minn., Mr. Gustafson added that “you just can’t kill” Town Cars, which he affectionately referred to as “BarcaLoungers on wheels.”

Pat Corbett, 70, is an industrial consulting engineer who lives in Manchaca, a suburb of Austin, Texas, and is active in his local Lincoln car club. His wife drives a 2004 Town Car. When his daughter, now 20, needed transportation in high school, Mr. Corbett gave her the ’91 Town Car he was driving and bought a ’94 from a seller in California, sight unseen.

“I put some new tires on it and drove it back to Texas without any concern at all,” he said.

Explaining that he favors the “pure mass” of a large Lincoln over airbags, Mr. Corbett said of his fellow Town Car aficionados, “We’re old enough to come from an era when driving someplace was an event. We weren’t concerned about trying to get from Point A to Point B as cheaply as possible. We don’t care about a 40-mile-per-gallon car that’s so small that you could put it in the trunk of my Lincoln.”

If you want to buy a used Town Car, a newer one in good condition will generally set you back anywhere from $4,000 to $6,000. Both Mr. Gustafson and Mr. Corbett think that they have the potential to become collectibles, and therefore more valuable.

Jerry Seibert, 68, a fellow Lincoln club member who lives in Springfield, Ill., is also bullish on the vehicle. “Every time I take my 2004 Town Car in for service,” he said, “I get a call saying that their general manager wants to make an offer on it.”