SAN FRANCISCO -- The Angels went into Sunday with a chance to secure a three-game sweep of the Giants and a winning road trip against two teams in the mix in the National League Wild Card race.

Instead, by the time the bottom of the eighth rolled around, infielder Luis Guillorme was on the mound to get the final three outs for the Halos in a 13-6 loss in the series finale at Oracle Park.

The Angels came out on the wrong side of dueling bullpen games, an unplanned approach for the Giants because scheduled starter Kyle Harrison landed on the injured list prior to the game. But San Francisco took a commanding lead early on and its 'pen held the line, limiting Los Angeles to two runs through the first eight innings before a four-run rally against closer Camilo Doval in the ninth.

"We just couldn't stop them, period, today," manager Ron Washington said. "You've got to give them credit for swinging the bats."

Rolling with a bullpen game was always going to be shaky for the Angels, whose 5.50 reliever ERA ranks 29th in the Majors. But Sunday underscored the team's need to find a better solution to fill the rotation spot previously held by Reid Detmers, who was optioned to Triple-A Salt Lake after his June 1 start.

The Angels were able to use an off-day to skip Detmers' spot the first time it came up, and they tabbed José Suarez the next time around. It did not go particularly well, as he allowed four runs across 2 2/3 innings in last Tuesday's loss to the D-backs.

The club tried something different on Sunday by having Suarez follow an opener. Hard-throwing rookie Ben Joyce pitched two scoreless innings and left the game with a 2-0 Angels lead, thanks to Kevin Pillar's RBI single in the first inning and Nolan Schanuel's solo homer in the second.

Then came Suarez, who worked a scoreless third inning after Luis Rengifo ran down Giants shortstop Brett Wisely between third base and home for the first out of the frame. Rengifo went down after that play with some apparent pain in his left leg -- he had to lunge to apply the tag -- but stayed in until the fourth, when he took a sharply hit grounder off his glove hand and exited with a left wrist contusion. He is considered day to day.

The Angels' defense could not bail Suarez out after that. The 26-year-old left-hander did not record an out in the fourth inning, which spiraled into a nine-run outburst for the Giants. Five of those runs, all earned, were charged to Suarez, while Hunter Strickland was tagged for the other four (three earned due to a Cole Tucker error).

Seven of the 10 batters Suarez faced on Sunday reached base. There were five hits (four doubles), one walk and one hit batsman. Suarez, whose ERA ticked up to 8.15, later said he believed most of the quality contact against him came on pitches that were out of the zone, but only one of the hits he gave up -- Thairo Estrada's two-run double -- was on a pitch that wasn't a clear-cut strike.

"I feel like I made pretty good pitches," Suarez said through interpreter Manny Del Campo. "They put good contact on it, and I can't control that."

After the upcoming three-game series against the Brewers, the Angels have two off-days bookending the Dodger Stadium edition of the Freeway Series, which gives them some flexibility in lining up their rotation.

One reinforcement is nearing a return from the 60-day IL, as Chase Silseth (right elbow inflammation) continued his rehab assignment with Salt Lake on Sunday and pitched five innings on 78 pitches. Before the game, Washington said Silseth could potentially be activated following this latest start, but it depends on how he recovers from it.

Sam Bachman could be another option, once he's eligible to be recalled. He was activated from the 60-day IL and optioned to Double-A Rocket City on Friday but was not as built up, most recently pitching three innings on 50 pitches on Thursday.

Asked if the team would continue to give Suarez opportunities in the open rotation spot, Washington was noncommittal.

"We haven't discussed that yet," he said. "I haven't had a chance to talk to anyone since the game ended. I'm not going to stand out here and try to speculate, so we've just got to wait and see where it goes."