The phrase "last leg of the race" is correct and usable in written English. You can use it when talking about a figurative race that has multiple steps or phases, in which the last phase is of particularly importance. For example: "We are now entering the last leg of the race for the Senate seat, so every vote counts.".
It was a thrilling last leg of the race.
The last leg of the race gives every indication of being very close, and Obama's ground game will bear the full weight of the wager.
The last leg of the race took runners on a bridge over the Potomac River, with no shelter from the sun.
We caught up only to have the top of our mainsail rip on the last leg of the race.
The final leg of the race was freestyle.
Instead, Scheidies will most likely walk the final leg of the race.
The ninth and final leg of the race to Kiel, Germany, starts June 8.
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MA of Applied Linguistic, Maquarie University, Australia