Whereabouts are you? Is there a local park where you can take your DS, and chat to other mums (or are they ignoring their progeny and other mums, and chatting amongst themselves?). Our local library does singalong sessions, which are good occasions to become familiar with local faces.
Your local council and Children's Centre may publish What's On listings, for stay-and-play sessions and playgroups that you don't need a "written invitation" for. FWIW, that kind of arrangement has "gatekeeper" written all over it, and is very rude, and extremely unhelpful to new mothers. If there is a number limit on the group, let them say that, not make a "written invitation"!
However, thankfully, "gatekeepers" of that kind don't necessarily represent all the mums they organise the group for, and there should be reasonable sorts around.
It only takes one acquaintance to start increasing your social circle. The only trouble is making that one acquaintance, as not everyone will do introductions (too lazy/preoccupied/shy/doesn't realise this is what you crave).
However, this is how is can work once you have met one or more people who like introducing people to one another:
I went to new playgroup last week on the recommendation of a mother I know from gymnastics and DS's nursery (that was why we first started talking), and brought another mum friend whom I also met through a mutual friend. I introduced C to S, and was introduced to H. This is quite something, given that my DS is now 2.6, and most of my original "grojup" has gone back to work, with a few exceptions who are off work again with Baby Number 2.
If you're anywhere near SE London, send me a PM! 