In your shelter/rescue admin area, on your My Shelter/Rescue Info page, you can select Act Quickly and that will display a red banner on all your pets "Needs Home Quickly". Screenshot examples below.
This is an account-wide setting, so all your pets will display it. Change the setting here:
https://www.adoptapet.com/shelter/shelter_edit_form
In the right column, look for
Does your organization euthanize to
make room for new pets coming in?
(Not shared with public, for our records only.)
Select yes. Then it expands to ask, "Do you want the words "Act Quickly!" to appear under your pets' photos on search results pages..." option.
Select yes.
Scroll down and click the Update Shelter/Rescue Info button.
You can always change it back, the change is instant. See how it looks by viewing your pets from any link at https://www.adoptapet.com/shelter/listing_links
The red banner also shows on pets in search results when adopters search for pets from www.adoptapet.com
Partial screenshots visual examples
On a search results page:
When you click on a pet:
Adopters who are especially passionate can even filter their searches to show only those pets: