Rather than waste trees by purchasing gift wrap made from virgin paper (often coated with plastic foil that can’t be recycled, and wrapped in more plastic), switch to recycled gift wrap paper instead. You can now find lots of lovely designs, if you choose to use disposable wrapping paper. All these selections are printed on post-consumer paper and come in zero waste packaging. So at end of use, either carefully unwrap and use again, or recycle easily. Most are sold in packs of folded paper (for an extra charge, most will send by the roll). Use with paper packaging tape and recycled gift tags.
Curlicue (UK) offers lots of lovely recycled gift wrap designs, with matching gift tags. Printed locally with eco inks, the range is illustrated with designs from favourite artists.