Action Overwrites JPEG file after the first copy

Another option is to record the Image Processor Pro script into an action. When configured similar to below, each use of the action will produce a new saved file with an incremented filename suffix:

Here is a script for this task (not my code):

A filename suffix will be added, such as:

_001.jpg 

_002.jpg 

etc.

EDIT: I just wrapped the original code posting into a try+catch for an open/saved doc and removed the original try/catch for a saved file.

/* Start Open/Saved Document Error Check - Part A: Try */
savedDoc();
function savedDoc() {
  try {
    app.activeDocument.path;
/* Finish Open/Saved Document Error Check - Part A: Try */

/* Main Code Start */
// https://forums.adobe.com/message/4453915#4453915
main();
function main(){
if(!documents.length) return;
var Name = app.activeDocument.name.replace(/\.[^\.]+$/, '');
var savePath = activeDocument.path;
var fileList= savePath.getFiles(Name +"*.jpg").sort().reverse();
var Suffix = 0;
if(fileList.length){
    Suffix = Number(fileList[0].name.replace(/\.[^\.]+$/, '').match(/\d+$/));
}
Suffix = zeroPad(Suffix + 1, 3);
var saveFile = File(savePath + "/" + Name + "_" + Suffix + ".jpg");
SaveJPEG(saveFile, 12);
}
function SaveJPEG(saveFile, jpegQuality){
jpgSaveOptions = new JPEGSaveOptions();
jpgSaveOptions.embedColorProfile = true;
jpgSaveOptions.formatOptions = FormatOptions.STANDARDBASELINE;
jpgSaveOptions.matte = MatteType.NONE;
jpgSaveOptions.quality = jpegQuality; 
activeDocument.saveAs(saveFile, jpgSaveOptions, true,Extension.LOWERCASE);
}
function zeroPad(n, s) { 
   n = n.toString(); 
   while (n.length < s)  n = '0' + n; 
   return n; 
}
/* Main Code Finish */
}

/* Start Open/Saved Document Error Check - Part B: Catch */
  catch (err) {
    alert("An image must be open and saved before running this script!");
  }
}
/* Finish Open/Saved Document Error Check - Part B: Catch */

https://prepression.blogspot.com/2017/11/downloading-and-installing-adobe-scripts.html

If you look at the File > Automate > Batch command, there is an option to override action save as names, as this is a common issue when batch processing saves from an action.

You are not using a batch process, so another way around the issue is needed.

As c.pfaffenbichler notes, a script would be required to work around the action naming overwriting.

If you want to save numbered copies that sounds like a task for a Script.