The short answer is - yes, there are a bunch of ways to be legally in the United States without a green card. These tend to be temporary, or contingent on other things, so harder to track and produce immediate evidence of.
The longer answer is, it doesn't matter. Deportation is a tool in the government's tool kit for dealing with those who entered the country illegally - but Deportation is a process, with steps, due process and it requires the ability for the potential deportee to argue their case to an immigration judge. Being grabbed by masked thugs off the street, taking a brief layover in Louisiana while they fuel the plane and then being deported to a torture facility in El Salvador without so much as ever speaking to your lawyer is a contravention of your rights. Your human rights, your 5th Amendment rights, take your pick.
Kilmar Abrego Garcia, for example, was never formally granted asylum. He missed the filing deadline of one year. However, in his original deportation trial he showed significant evidence and credible testimony that he had fled El Salvador to escape being forced to join the gang that was shaking down his family for protection money. The only reason he wasn't deported after that original hearing was that a Judge granted him hold order - stating he specifically could not be deported to El Salvador because his life would be in danger. It's unlikely he has any simple card or document that shows that order he could carry with him. In direct violation of that order, ICE sent him anyway. He's almost certainly dead.
I'm an old man, but I'm 90% sure I'm saying this right: Fucking Based.
"What gender are you?"
"Lovecraftian nightmare tentacles"
"But what pronouns do you use?"
"It/Insanity Screeching"
"Hmm, I'm starting to think we mortals might be using restrictive definitions of shit"