Helping rescues helps rescues help dogs, but I do understand what you mean. It's nice to be able to help specific dogs that speak to your heart.
I think if you want to help facilitate the rescue of a particular dog and you're not close enough to help with transport, most of your helping options are going to be limited to crossposting the dog's information - which, if it's already on here, is not going to be as necessary, and financial support - which is ALWAYS welcome.
Offering to sponsor the dog - pay the pull fee, or for their vetting, or to board them, can all help enormously. You can also call the shelter and offer to sponsor the dog and they can then list the the animal as "sponsored" on their website.
However, a quick word about sponsorships: I know some people may disagree with me, but I am of the opinion that sponsorships should be offered to approved (reputable) rescues ONLY. I do not believe in paying a dog's adoption fee so that it can be adopted locally for free. You are not doing that dog any favors. If someone cannot afford the nominal adoption fees that most southern shelters charge, or is choosing a dog simply because it's free, these are not placements you want to facilitate.
ETA: Sponsorships are wonderful, but if you go that route, do remember that the limiting factor for most rescue groups is not money, it's foster homes. We all need money, don't get me wrong, but having places to put all the deserving dogs is the harder part of the equation. Even with sponsorships, rescue groups may not be able to take a dog - also a group that will take anything, especially anything sponsored is a red flag that they're turning over dogs for profit and again, not something you want to facilitate.
What about becoming a foster for GSRNE? And then offering to foster the dogs that appeal to you off the Urgent page?