Can't please everyone, feeling himself part of the industry, try the hardest, helping to rehome these dogs? Sure there is an organisation near you, who's Greyhounds you could promote. And if you do not want to ask your neighbour, maybe promote those of a bit further away?
It is not fair to these former racers, to simply forget about their needs, if we got out of them, what we wanted.

Do them the favour to promote them as companion/pet dogs to get them out of rescues and kennels, please. Who could know of their benefits, if not the Greyhound people?

.. just my thoughts anyway please ..

It might help with promoting, to walk a single or a pair of your Greys outside. Someone you meet may dare to talk about your dog if it's a single one or two. Conversation is the aim of this walk, by the side it's training ;) Sure there will come a lot of wrong suggestions of all kind. And I am sure it's not just fun. One will get an idea what people out and about think of the whole Greyhound topic. If you walk a retired one, this may be an oportunity to meet a new owner as well. Always the question, if one is able to get in touch with the right people and get this relation built up.
A lot of crap ideas out there about the industrie, greyhounds, cruelty and so on. All facts(!), but much less regular than generally thought within the society. This is probably only be cured in thousands of direct conversations. They can not be payed - such talks need to be hold and a well-mannered Greyhound by the side will make it probably more easy.

My second point
is one of the most easy for trainers and staff:

To name the temper of the single dog/bitch for rehoming thorough.

See, there are hundreds of black Greyhounds in kennels, looking for a permanent pet home. And they are not getting less if i value the winning hounds right, most are black. So breeding will go further the black direction. They look so similar one to the next on photos, that the discription of the charakter needs much more weight.
Only "lovely" or "sweet" "well-mannered" alone might not help this one dog finding the right match.
But as a Trainer, Owner, Staff, who has professionally worked with the dog, you know exactly what his favs are and what he doesn't like. What it's plus and minus are. You can tell a whole universe about the single dog making it unique. And that is what all these black Greyhounds in kennels lack: Beeing recognised as unique and worth taken home as pet after their racing days. Just solve it....

I've heard the argument money to often, I think, although I can't see why beting industry is allowed to leave that chapter to owners while making money out of the dogs. And kennels as photos for a rehomable hound are vital, the better they are, the better are the chances. Often rehoming feels a competition like in racing.