L.H.
M.,
this is a little different than just a "play date" but our church has a women's daytime Bible study on Wednesday mornings that should start up again next month (don't have the exact week yet, but I can let you know). It provides childcare and there would be some kids your daughter's age and a bunch of other women you can be intentionally getting to know on a deep level.
Here's the website: www.bygraceonline.org. You are also, of course, obviously welcome to come to anything else before then! We've got a growing number of stay-at-home moms and always enjoy the Wed. morning times together! (we are not all moms who meet, but most)
As for a completely different approach, if you are anywhere near-ish Newport News, you can check out the Riverview Gymnastics Center that offers dance and gymnastics classes on a quarterly basis (for a good price). My girls have made friends that way (as have I!). There are also other types of classes at various locations around the city. You can check it out at: http://www.nngov.com/parks-and-recreation. If you aren't in Newport News, you can look at your local city's parks and recreation department to see if there is something offered closer to you.
Feel free to message me if you want further ideas or info, or if you want to connect! And welcome to the area!