I'm trying to learn more about MPLS-TE, specifically how packets are routed once the TE LSP is set up.

Most of the examples I could find that include the MPLS label stack shown are examples for Fast Reroute, e.g. link/node protection. For example, if we have Router A -> Router B with the label 10 being pushed to get the packet from A -> B, with the protection LSP being A -> C -> B, when the A -> B link fails, we have another label, say 20, pushed on top of the label 10, so that traffic now goes through the protected LSP A -> C -> B.

Do all TE routes work this way? Does the original LSP path (w/o TE) have to be only one hop away? This is the part that confuses me - if it's multiple hops away, how would the head-end know which label to push to get to the tail-end after the packet exits the TE LSP?

For example, if my original LSP is A -> B -> C -> D -> E, and I have TE LSPs B -> F -> D and B -> G -> D, how would B know to push which LSP label onto the packet before it enters either TE LSPs so that when it gets to D, it can continue to E? There are 2 labels, the top one being the TE label, and the bottom one being the LSP label, right?