Since you bought separate tickets (whether or not from different websites) this is not a misconnection. How you are accommodated should you reschedule or should you miss the "connecting" flight depends on the grace of God (or the airlines concerned). The airline that made the schedule change should allow a free switch to a flight that works better for you.

If you have to book legs separately, you would have much more peace of mind if you can allow for an overnight or two at the city in the middle.