Is this possible in AI or any plugin?

THE-KAL,

With the native tools you can try this approach (remember to deselect in between Direct Seletion Tool operations):

1) ClickDrag to stretch, as you did,

2) ClickDrag the Handles at the inner ends of the stretched path segments downwards to fit (the bottom one first), maybe also one or both of the Handles at the far end.

For more control, before 1) you can add Anchor Points in one of two ways, depending on desired appearance:

Either A) (roughly) where the segments are horizontal (some fonts have them there to start with), and also select them before a first ShiftClickDragging in 1) (using Shift to move the bottommost part horizontally and keep the height/depth of the letter), followed by a ClickDragging of the end Anchor Points to their desired end position.

Or B) (roughly) where the segment shapes are to be preserved (closer to the inner ends where they are still sloping) to lock thé inner ends, then just ClickDrag the the Anchor Points to their desired end position.

Corresponding to 2) you can adjust any relevant Handles as needed; there is a way to do that fully controlled

I am aware of extension tools, but they don't help much in my case.

Astute Graphics' VectorScribe plugin includes a path extension tool. The tool can be hit or miss at delivering desired results on curved path segments. The tool does at least display a preview line of where the path would extend. Sometimes that preview line will bend in some very odd directions.

You could check out the Reshape tool https://helpx.adobe.com/illustrator/using/adjust-path-segments.html#stretch-parts-of-a-path

There is also a tool for continuing paths in the Astute Graphics plugin suite.