The Amazing Spider-Man has been extremely popular, both with audiences and the box office. The remake of the Sam Raimi original, grossed well over it's expected profit margin and is now confirmed for at least two more sequels with Andrew Garfield starring. Speculations have no arisen that Sony may take full advantage of the success and possibly create spin-offs with different characters but set in the same world as the web-slinger.
According to Total Film Sony's Co-Chairman said, “We are going to access Marvel’s full world of Spider-Man characters, so be on the lookout for new heroes and villains.” With that being said, many are saying a Venom film could happen, or maybe even Black Cat. With a universe as wide and varied as Spider-Man it's hard to speculate on just one character.
Michael Lynton told Deadline that they are working very closely with Disney and Marvel, since Sony still technically owns the rights to Spider-Man but is property of Marvel. Even with the new films, the studio is still thinking of expanding the Spider-Man universe in other places, but that's for the audience to wait for.