Pull out deleted handle without disturbing active

LoneLeon,

To give a full solution, I worked on a demonstration of how to add a new Handle at another angle while keeping the active Handle thereby creating a corner Anchor Point between the two curved segments.

But based on a reinterpretation of your question to mean that you actually wish to recreate the original opposite Handle, which is much simpler, with the added option of changing the new curvature, and recreating a smooth Anchor Point, here is a full solution, Smart Guides being your friends, telling when you are within snapping distance:

1) This is just a crude sample with a deleted Handle at the bottom corner;

2) ClickDrag with the Line Segment Tool from the active Handle into the Anchor Point to create a guide line shown in red;

3) With the (normal) Selection Tool Alt/OptionClickDrag the guide line by its Handle end Anchor Point to snap to its Anchor Point end (Smart Guides say anchor); holding Alt/Option creates a copy guide line shown in blue;

4) With the Convert Anchor Point Tool ClickDrag the from the Anchor Point along the blue guide line to snap to its end, or if this creates reversed Handles along the red guide line to snap to its end instead, this depends on the direction of the path; either way, this will restore the active Handle and recreate the deleted Handle, in other words bring you back to where you started;

5) If you wish to have a different curvature/Handle length for the restored Handle, deselect (click an empty spot) with the Direct Selection Tool, then select the segment with the recreated deleted Handle (by clicking it), then ClickDrag the Handle from its current position along the blue guide line to the position that gives the desired curvature/Handle length shown in yellow (Smart Guides say path); if you wish to make it longer instead, start by extending the blue guide line using the Transform panel;

6) This just shows the paths created in 4) and 5) without the guide lines for easy comparison with 1).

Click to get closer, Click again to get closer still