The enemy criminal is captured by the player, and a middle-ranking one of an enormous organization at that. There's no way he didn't know the high-rankings, otherwise he couldn't coordinate with the lower ones. As a middle-rank, he must also know the mundane, day-to-day stuff or he wouldn't be able to organize his Crime Cell.
By my—the DM's—roleplaying, he is not someone that would easily leak information. As a field leader of his smaller Crime Cell, he had the experience of being dead at least three times and are trained in anti-torture training. He constantly talks back, goad the PCs to emotional, not caring if he's going to be killed or not, because at some point the organization will sent an evil cleric to revive him in the future, like in the past. (The players killed him before I managed to hint why he had this uncaring attitude.)
Starting at 6th level, you can curse the soul of a person you slay, temporarily binding it in your service. When you slay a humanoid, you can cause its spirit to rise from its corpse as a specter. When the specter appears, it gains temporary hit points equal to half your warlock level. Roll initiative for the specter, which has its own turns. It obeys your verbal commands, and it gains a special bonus to its attack rolls equal to your Charisma modifier (minimum of +0). The specter remains in your service until the end of your next long rest, at which point it vanishes to the afterlife.
The Hexblade then asks the specter if he can understand her. As per Specter's monster stats, Specter do understand the language, just can't speak. The Hexblade then orders the Specter to answer her questions with "Yes" by nodding, and "No" by shaking its head.
Because of "The Specter remains in your service until the end of your next long rest", that means the Specter can be continually asked for a long time. Meaning there would be a lot of fundamental things that they can ask, if they had the right idea. Also, it had advantage over Speak with Dead because they are compelled to 'answer' truthfully when ordered.
Now, for this particular enemy I'm not too worried because his area of influence are only the capital where the PCs have transit, and orders are sent through Sending. But what about in the future where they managed to snag a high rank? The Organization is thousand-year old criminal group, and they even had a rulebook for their regular guards about To-do and Not-to-do that are based on their own ingenuity of slipping through guards, dozens of evil cleric capable of revival, etc etc. This Accursed Specter seem to be a kind of weak spot they wouldn't normally left open.
Should this be allowed?
(I already okayed it by "That is an excellent creativity. For this time, alright. It's both as-written and doesn't fully overlap with Speak with Dead. But I'll ask around later in case you want to do it again. There might be some rule or something I didn't remember prohibiting this")