When you scan a previously screened image the existing halftone screen is visible.
1) Try angling the image on the scanner bed to reduce the moiré effect as much as possible
2) Run the despeckle filter on the image
3) Let me recommend an extreme measure which may work for you.
Scan the halftoned image at a resolution to yield approximately 4 pixels per halftone cell. (IOW, if the image is screened at 150 lpi, scan at 600 dpi.)
Run the Median filter with a pixel radius of 3 (IOW, smaller than the individual halftone cell).
Run Unsharp Mask at 250% (or more), with a radius of 1.5, and a threshold of between 12 and 18 levels.
You might want to play with a limited selection in order to see what combination of settings you really want. The Median filter will blur your image and the point of the USM is to get some of the focus back.
Unfortunately, you will lose some detail and more subtle tonal variations will tend to flatten out. The trick is to avoid getting the picture looking like a painting, (unless, of course, that effect is acceptable, in which case, wail). Posterization is a problem with extreme settings.
(Lurkers: This is an if-all-else-fails measure. Warning: Professional driver on a closed track; do not try this in your car. Objects in mirror are larger/closer than they appear. Your mileage may vary. Taxes, title, dealer prep, and options not included.)
Also; you might try rotating the original on the scanner glass and seeing if you can cut down on the moire that way. Sometimes, esp with camera seps or halftones, the image is not screened with a true horizontal origin and you may abe ADDING to the moire.
It is *much* easier to remove moire from colour images than B&W, in so much that there are 4 half-tone dots (to form the rosette) for every one dot in Monochrome printing, and consequently the screen is effectively 4x as course for B&W.
Check first to see if your scanner software has a Descreen setting. - Some high-end scanner software such as FotoLook for AGFA scanners has this option with variable LPI settings. Other cheaper scanners has build-in descreening, in which case you wont ever see noticable screening.