If you want to get into leveraging Virgin Atlantic (VS) as a transfer partner, the absolute best thing you can do is learn how to manipulate the web address of Virgin’s Awatd Seat checker page to run searches for award inventory on their partners.
This FM article I’ve posted below shows how to do this for Delta flights, but the trick works for other Virgin partners like KLM and Air France as well. You will need to know what routes each partner flies directly to return correct search results, so it’s not as easy as it might first seem. If the award is not showing on a particular day via this checker, though, VS most likely doesn’t have access to any more award space on that flight / those flights on a particular day. Not all seats are available to be booked as awards, and not all award seats are available to partners.
If you want something that’s a little more user-friendly, try an award booking tool like seat.aero to look at a whole bunch of dates at once. You’ll probably have to pay some money to get the best experience, but it’s a small fee compared to the power you get from being able to search such a wide range of dates, routes, and airlines all at once.
A couple of other pieces of advice: if you’re playing with Virgin, avoid flying through London as you’ll get destroyed with taxes. You also have to be pretty flexible with dates if you want the three best award booking prices. If you’re trying to fly on peak days of the week at peak times of the year, it’s going to be highly unlikely you’re going to find any availability.