The fifth-seeded Swede was in deep trouble as Hewitt took the first two sets. The Australian pulled off a spectacular diving backhand winner to break for 5-3 in the second set.
Soderling used his reliable serve to work his way back into the match, which was played under the Centre Court roof because of the threat of rain. He broke in the last game of each of the last three sets to seal victory.
When Hewitt netted a forehand to lose his serve at love in the final game, Soderling sank to his knees and clenched his fists.
"A player like Hewitt is not easy and I really showed I can play well against a good player on the big courts and turn around matches like this," Soderling said.
Soderling, a two-time French Open finalist, reached the quarterfinals at Wimbledon for the first time last year, losing to eventual champion Rafael Nadal.
Del Potro took his shoes off after Wednesday's match was suspended by darkness after the first set and threw one out of the stadium and the other one deep into the crowd.
When he came back onto Court 2 on Thursday, he dominated by winning 12 of the first 13 games.
Del Potro won the 2009 U.S. Open but has struggled with injuries.
The second-seeded Serb has won two straight since his 43-match winning streak was ended by Roger Federer in the French Open semifinals.
Djokovic has twice reached the semifinals at the All England Club, in 2007 and '10. He can earn the No. 1 ranking if he makes the final this year.
"On the important moments, I served well, I played well and I returned really, really well today," Djokovic said. "In the first week, you have to stay focused. You're playing lower-ranked players with nothing to lose. It's just great to win two straight-set matches."
Federer wasted no time in reaching the third round at Wimbledon, beating Adrian Mannarino of France 6-2, 6-3, 6-2.
The players were kept waiting in the locker room by the two preceding matches on Centre Court that both went the distance, but a dominant display by Federer ensured it wouldn't be a late night under the roof.
The six-time champion broke five times to win in 1 hour, 28 minutes, finishing off the match with an emphatic smash.
Federer will meet 2002 Wimbledon finalist David Nalbandian in the third round.
Almagro won 7-6 (3), 7-6 (5), 6-7 (5), 6-3, despite Isner's 28 aces on Court 18. That was the site of Isner's record-breaking 11-hour, 5-minute victory over Nicolas Mahut in the first round at the All England Club in 2010, a match that ended 70-68 in the fifth set.
"It's the same as any other court," Isner said, "but it was a bit weird out there."
In a rematch of the longest match, Isner beat Mahut in the first round this year, too.
But against Almagro, Isner never earned a single break point all match. Almagro earned the only service break to go ahead 2-0 in the fourth set.
Almagro had 63 winners and only 14 unforced errors. He finished with 25 aces.
"It felt like he matched me, serve for serve," Isner said. "I knew going into the match that he had a really good serve and was going to get a lot of free points. But I didn't really expect that, to be honest."
Isner fell to 11-14 in 2011, and the best he's done at any tournament is reaching the quarterfinals at Auckland and Houston.
His ranking has fallen from 19th at the start of the year to 47th currently, his worst spot since September 2009.
"It's disappointing, to be honest," Isner said. "It hasn't been a great year. It's tough, and I wish I knew exactly where and what went wrong. But I'll be fine. I'll find my way."
Also advancing Thursday were No. 12 Jo-Wilfried Tsonga, No. 18 Mikhail Youzhny, No. 19 Michael Llodra and No. 32 Marcos Baghdatis. No. 13 Viktor Troicki, No. 20 Florian Mayer and No. 26 Guillermo Garcia-Lopez were upset.
Information from The Associated Press was used in this report.