This is my spin on things as a pilot, and other pilots should reply also.
First, I only take the time to scan new posts and I look only at the subject line. That means if the request does not list the starting point plus the ending point I will not look at the thread. If it lists a starting point and an ending point such as Knoxville TN 37950 to Jamestown NY 14701 I will look further. If it lists a start and end point that does not list my home town of Knoxville, but crosses over Knoxville I will also look. The number of miles then comes into play.
I will not fly any more miles than a comfortable range. I have a plane able to cover greater distances than most, but when a transport is longer than 500 miles I pass on it. For most pilots it should be 300 miles. But what if you live in Alabama and need to get a transport to New England? You have a couple of choices. Hope you can get a group of pilots to split your transport into a relay and make it work (very difficult) or search for intermediate stops along the route to foster the animals and treat the transport as distinct and separate transports. You will have an easier time getting pilots to fly a 300 mile or less transport on their schedule, rather than hoping two or three or more can coordinate schedules to do a long run.
Leave the word "Urgent" or "must transport this weekend" from the subject line. As a pilot that shows you have no flexibility and I don't know many pilots that want the pressure of having to get a transport done when the potential for bad weather or other factors influencing the safety of flight exist.
Finally, be proactive. Use the maps on the web site to find pilots along the route and send them a personal request. Show a willingness to adjust your schedule or start or end points to meet their needs. As pilots we all have different hot buttons. I'm OK with weather and slightly longer distances, but i will avoid flying a single animal unless there is a very special need. I only have so many dollars and I have determined I can make a bigger impact by filling my plane. Others are very comfortable flying a single pup, but may have weather limitations. I mention this because all sorts of variations may exist that are not readily apparent so you need to be a little flexible. If you only have a single pup to transport, but you also know of another transport parallel to the route, or overflying the route with other animals try to combine the two. Or if the pilot only wants to carry one animal, try to make it two transports.
So the key is posting the correct subject line, and maintaining a certain flexibility. Keep the transports to reasonable distances. A single pup that needs a 700 mile transport is using a huge amount of resources and money. The pilots may be reluctant to spend $1000 to transport a single pup. Lastly, as you continue to do transports develop a relationship with the pilots. Get to know them. They really want to help if they can tailor what they do to their own specific capabilies.