Let's be honest: it's very difficult for PCs to die in OSR+.
While combat in the game is fairly deadly (there's low HP and fixed damage to match), the players have a lot of narrative "outs" in the form of story tags and fate points. Stabilizing characters on death's door is not difficult unless the player already got themselves into lots of trouble, as the TN only increases if the character is wounded.
That being said, it's possible to lose a character to a chaotic battle (a PC goes down and no one can get to them in time to stabilize them, and the player who controls that PC is out of fate points). OSR+ does not have resurrection as a feature in the game, so when a character dies, they're dead permanently. Moreover, players might decide that this is the ideal moment to let their character pass through those pearly gates by submitting themselves willingly to an insurmountable challenge.
If a PC dies mid-session, let the player assume control over other NPCs in the party or even GM-facing NPCs (such as other enemies in an encounter).
Introducing a New Character
Once the player has time to think about how their new hero will get introduced into the game, you can run a session zero with just that player and one other player in the party, bonding the new PC to the existing party member.
Like any other PC, we need story hooks to get the new PC enmeshed with the narrative as it exists. Since you'll likely already have adventure hooks written into scenes that tie into PCs' story hooks, this is an opportunity to marry the new PCs' conflict with that of the other PCs.
Beware Lone Wolves
You don't want to introduce a new PC that has a completely different agenda than the party's, or that will result in your having to radically alter your adventure web to accommodate discordant goals: instead, you should feel empowered to steer the player into making choices that are compatible with the narrative thus far. In short: it should be a lot easier to bring a new PC into an existing campaign than creating one in a vacuum!
If you’re going into a one-shot where there’s a high likelihood of dying (Ride the Lightning comes to mind: a high-octane PC meat grinder involving battle trains), you might want to encourage players to develop backup heroes that can be re-inserted into the action immediately. This of course only makes sense if your game allows for it, as shorts and longer campaigns necessitate a session zero to ensure that the new hero comes in with story hooks to properly integrate them into the existing narrative.