The British were prepared to give it all to the Americans, but Polk quickly accepted OTL borders because he was getting ready to fight Mexico and didn't want to be distracted.
If anything, if the US is fighting a war then the British will hold any deal. Polk was under a time limit. Either get the Brits to give in earlier or hold off the Mexican war.
Or fight the Mexican War sooner. In 1836, when the new Republic of Texas requests annexation, have Congress accept it, leading to the war happening then, instead of 1846-48. This also results in Americans settling the Pacific Coast earlier, so that there is a larger American presence in the territory by the mid-1840s.