Android 4.x and earlier don't support modern TLS by default

While testing an Android app right before release, I found that on old versions of Android, it failed to complete any web requests to my server. The Volley error response indicated that the cause was an SSLException, "connection closed by peer." Apparently the only TLS version supported by Android 4.x is TLS v1.0, which was disabled on my server a while back. Google Play Services has a provider for modern TLS from which an SSL engine can be created, but I'm not yet sure how to make this work with Volley.