Okay, so I did some research and dug up some info on the whole "Ringo's secret marriage and second baby" rumor thing.
Here's the Japanese source that's being quoted. It's from Yahoo Japan, but sort of like a side (gossip?) magazine, so I don't think we can trust it 100%, but damn if it doesn't have some very plausible stuff in it.
Basically, an alleged neighbor of Ringo's has spilled a bunch of dirt on her. According to this she DOES live in Saitama (as I thought based on her comments in an interview). It says that for the past two years, she has been dating Yuichi Kodama, the director of some of Jihen's music videos (this is from a "music industry insider"). They are half-living together and in a common-law marriage (note: the rumor was that she had a secret marriage. According to this article, the marriage is not legally registered).
A reporter (or the neighbor?) apparently saw Ringo, Yuichi, Ringo's son, and a baby go into an Italian restaurant in Saitama together towards the end of August. (There is a pap photo of this; I've seen it but I'm not sure where it is.) The baby looked to be about 4 months old "and the eyebrows resembled the older son's". Later the reporter accosted Ringo and said "Congratulations on the birth of your second child." She said "You've got the wrong person" and continued to insist that repeatedly but without trying to hide her face.
Honestly, I think this is likely almost 100% true. Her current appearance is with a few extra pounds - the most/only plausible explanation in Japanese society for female singers is that this is post-pregnancy weight. I know she lives in Saitama. The family demographics all match up. She wanted to be married again, she said that in an interview (even if this isn't an official marriage).
This could even explain why Jihen ended. She wanted to have another kid and focus on her family.
This also explains why all the tour dates are in Tokyo, which is actually kind of strange if you think about it - 7 dates and not ONE is in Osaka or something?
If the baby was 4 months at the end of August, he/she was born around April/May. So, she was pregnant from around June/July 2012.