Lex,
As I (mis)understand it, the best way would be to:
1) Establish which font it is and use that if possible, or use one that is close enough for you.
You can try the Adobe Fonts site (upload an image, clicking to the right in the search box, to find the same or similar) and have immediate access to your find(s),
https://fonts.adobe.com/
or you can use Photoshop to find the same or similar Adobe Font as described here,
https://helpx.adobe.com/in/photoshop/how-to/match-font-image.html
or you can try the following font finder sites,
https://www.whatfontis.com/
http://www.identifont.com/
https://new.myfonts.com/WhatTheFont/
https://www.fontsquirrel.com/matcherator
http://www.bowfinprintworks.com/ScriptIDGuide.html (Script fonts)
or hope that someone here knows it.
2) Write the new text once you have the/a font to work with.
3) Place the new text on top of the old outlined text and change width, height, and position, for best fit.
You can do that by changing the values in the W and H boxes Transform panel and move the new text round with the Arrow keys and/or ClickDrag with the Selection Tool and/or change the X and Y values.
If you outline the new text you can start by aligning it to the old one using the Align panel; it is very likely that it will be best to end up making adjustments based purely on visual appearance.