A hat with a wig attached is typically referred to as a **hat wig** or **wig hat**. This accessory combines the convenience of a hat with the look of hair. Here's how to describe it:

- **Hat with attached hair**: A hat, often like a baseball cap, beanie, or sun hat, that has synthetic or natural hair sewn or attached directly to it, giving the appearance of a full head of hair when worn.
  
- **Pre-styled hairpiece**: The wig is usually pre-styled, meaning it requires little to no effort to wear and style, offering an easy solution for covering hair loss or creating a quick, stylish look.

- **Comfortable and versatile**: These hats with wigs are designed for convenience, providing both sun protection and a natural-looking hairline without the need for a separate wig cap or hairpiece.

- **Common in cosplay or casual wear**: Wig hats are often used in costumes, but they can also be practical for casual use, especially by people looking for quick and hassle-free hair coverage.