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'GoT's Sam Needs To Hear About Jon Snow's Parents ASAP
There was a major piece of information disclosed to Sam Tarly during the Game of Thrones Season 7 episode "Eastwatch," courtesy of Gilly, and it seems that everyone except Sam made the connection. As frustrating as it was to see Sam totally ignore Gilly's revelation that High Septon Maynard annulled Prince Rhaegar's marriage and performed a secret wedding in Dorne — which would give Jon Snow a legitimate claim to the Iron Throne — there's a chance Sam could still learn about Jon Snow from Gilly again or he could stumble upon that fact on his own. Basically, don't give up hope that Sam will just completely miss out on this huge bombshell, even though he does deserve to be shaken by the shoulders for totally talking over Gilly at the most crucial time in the show.
All frustrations with Sam aside, however intense they may be, we should cut him a little bit of slack right now. He does seem really intent on finding a way to defeat the White Walkers that it's understandable he's steamroll over any information not directly relevant to the White Walkers right now. He want's to help his bestie Jon Snow, so the White Walkers would take precedent.
Additionally, I should note that Bran is really the only person (both living and relevant to this story in this show) that knows even a portion of Rhaegar and Lyanna's story as it relates to Jon. He's been so odd lately in his behavior that it's doubtful he'd send a raven to Jon Snow or Sam if he hasn't done so already. As such, we can't blame Sam for not immediately perking up at the mention of Rhaegar and an annulment. How would he have known that that information would be relevant — nay, crucial — to tell Jon Snow?
The best way for this information to get back to Sam would be Gilly, who would imaginably become so intrigued by High Septon Maynard's notes about this annulment that she would continue to natter on about them to the point that Sam would simply look at Maynard's notes himself. Or, Sam could read about the information independently and, fingers crossed, put the information together himself.
Basically, all hope isn't lost for Sam putting the clue together. It's annoying that he's sleeping on a real bombshell of a detail, but the way things are going in Season 7, there's not doubt the truth will out before the season finale.