Push mesh to meet new lines on same axis

I was hoping to understand the best way to achieve the following: to push this mesh to meet the edge line which will allow me to sculpt to the road area I’m hoping to extend to.

I was thinking cage edit, but it seems to resist actually pushing outwards to make a wider roadway.

Ultimately I’m trying to explore methods of freeform road modelling starting with an existing rough concept road.

Thank you