Chicken wire is quite strong, and available galvanized or coated in different colors of vinyl. Merchants sell rolls in several lengths to suit your needs.
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Start at one corner of the vinyl fence and unroll the chicken wire so that the end you are using comes off the roll next to the fence. For example, if you are starting at a left corner and working to the right, the roll of chicken wire should turn counterclockwise as it is unrolls. If you are starting at a right corner and working to the left, the roll should turn clockwise as it is unrolls.
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Fold the chicken wire around the first corner post of the vinyl fence so that the end is midway on the side, perpendicular to the side you are working on. Staple the chicken wire to the corner post, starting with the bottom edge at ground level. Continue stapling the wire upwards about every 3 inches to the top edge of the wire. Use galvanized staples, which won't rust and will hold the fence in place much longer than standard staples.
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Unroll the chicken wire past the next vinyl fence post and pull tightly by hand. Staple the wire to the post, starting at the bottom and working your way up to the top of the fencing while pulling it taught at each staple.
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Continue around the fence line in this manner.