The first two police officers to reach the suspect in Wednesday’s shooting rampage at the UPS distribution center in San Francisco almost didn’t make it to the scene because their squad car wouldn’t start.

It was a sequence right out of a Hollywood movie, with the officers having to commandeer a passing car — along with its driver — to respond to the “shots being fired” call.

It started just before 9 a.m. when two Mission Station officers, whose names haven’t been released, were walking the beat on Castro Street near Market. Their hip radios crackled orders to respond to reports of gunfire and multiple victims down at 17th and Utah streets.

According to a report later filed by one of the officers, they ran to their parked 2009 Ford Crown Victoria patrol car and turned the key “multiple times” — to no avail. The car would not start.

The pair jumped out and quickly spotted a young woman driving a blue Honda south on Castro. They flagged her down and told her they “needed to use her car to get to the incident,” the officer reported.

With that, the woman hopped into the back seat and away they all went.

As they neared Mission Station, the officers saw another parked patrol car. At that point, they thanked the young woman for her help, exited the car and jumped into the black-and-white.

Using a master key, they fired up the squad car and sped off.

Minutes later, they pulled up to the UPS building and made it inside to find the gunman had just shot himself.

With the suspect bleeding from the head, the pair provided emergency aid until fire department paramedics showed up and took over. The suspect, Jimmy Lam, 38, of San Francisco, was later pronounced dead at San Francisco General Hospital. Three of his UPS co-workers were found fatally shot, and two others wounded.

As for that car that wouldn’t start? The official line from the department is that someone left the car’s spotlight on, but the officers told their union reps that the car battery was dead when they started their shift that morning, so they used a portable jump starter to get it going.

They had driven around on their beat, making stops here and there, before the battery died again.

“It wasn’t the first time that officers had to use a box jumper to get it started,” Police Officers Association President Martin Halloran said.

He said the real issue is that San Francisco police have “one of the oldest fleets in the Bay Area, if not the state of California.”

The department itself estimates that nearly a third of its 800 patrol cars are older than 10 years, and another third of the fleet’s cars have clocked over 100,000 miles.

The 8-year-old Crown Vic that wouldn’t start had logged 107,000 miles. In contrast, Halloran said, most police departments sell their vehicles after three to five years, or once they hit 100,000 miles.

Just last week, a Board of Supervisors committee authorized buying 70 new marked and unmarked police cars in next year’s budget — that’s less than half the 185 requested. However, an additional 70 have been green-lighted for the following year.

“It’s no secret that we have some older vehicles, and we are trying to bring the fleet up to date by establishing a reserve,” Police Department spokesman David Stevenson said.

Still, Assistant Police Chief Toney Chaplin cautioned that age isn’t the only problem facing the department’s fleet.

Officers occasionally have had to jump-start even newer vehicles because radios, computers and overhead lights drain batteries.

“These cars have an enormous amount of technology that they did not have in years past,” Chaplin said.

Oakland high: Oakland Mayor Libby Schaaf was riding high atop her fire snail art car during Thursday’s big Warriors championship celebration near Lake Merritt — and with good reason.

In spite of a bumpy ride at City Hall — including turmoil in the Police Department, the Ghost Ship warehouse fire that killed 36 people and the Raiders’ move to Las Vegas — city voters still appear to be in her corner by a more than 2-to-1 ratio, according to two recent polls.

A Fairbank, Maslin, Maullin, Metz & Associates poll of 800 likely Oakland voters in late April found that 58 percent of those surveyed had a favorable opinion of Schaaf, while 25 percent viewed her negatively. The remaining 17 percent had no opinion.

A David Binder poll of 400 likely voters, conducted June 5-8, had nearly identical favorability numbers — but it also gave the mayor some reason for concern as she heads toward her campaign for re-election in November 2018.

Asked if the election were held today, 44 percent said they would vote to re-elect Schaaf — while 33 percent would vote to replace her. Twenty-one percent offered no opinion.

“Younger voters, African Americans, renters and newer voters are more likely to want to replace Schaaf,” Binder said in his analysis for the Jobs and Housing Coalition, an Oakland nonprofit business advocacy group that paid for the poll.

The poll also asked about a couple of names being mentioned as possible challengers — Councilwoman Rebecca Kaplan, whom Schaaf defeated in ranked-choice voting in 2014, and two-time contender Ignacio De La Fuente, for whom Schaaf worked when he was a councilman.

Kaplan had a 52 percent favorable to 18 percent unfavorable rating; De La Fuente’s numbers were 32 percent to 20 percent.

San Francisco Chronicle columnists Phillip Matier and Andrew Ross appear Sundays and Wednesdays. Matier can be seen on the KPIX TV morning and evening news. He can also be heard on KCBS radio Monday through Friday at 7:50 a.m. and 5:50 p.m. Got a tip? Call (415) 777-8815, or email matierandross@sfchronicle.com. Twitter: @matierandross