If your standing adjacent to a monster you can do a movement action, attack, then move. Movement points do not need to be spent before they are interrupted by another action - totally legal move.
The MP granted by Elven Boots is gained at the start of the hero's turn, but it doesn't need to be spent then. Like all MPs (including fatigue MPs, btw), you can gain them early and wait until later to spend them. The only catch is that they don't carry over to next turn, so if you end your turn without spending them, they're forfeit.
This is why heroes who are worried about Pit Trap can spend 1 fatigue to make sure they have an extra MP in the pool before the card gets dropped on them, but they CANNOT spend 1 fatigue to gain an MP in response to the card being played. If the unspent MP is already in your movement pool, you're good, but once the card is played it has to be resolved before anything else happens.
The only time that EB's "start of turn" timing would come into play is if there was effect that triggered when your hero gained an MP. Such an effect would need to be resolved during your hero's "start of turn." I can't think of any such effects off the top of my head, btw. All the ones I know trigger when the hero spends an MP, or moves into/out of an empty/adjacent space. Those effects obviously trigger when you're actually using the MP, which may be later in the turn.