Thyme is a "woody" perennial herb coming in hundreds of shapes, colors, and sizes. It grows flat and fuzzy to full and lush; it spreads, creeps and blooms with beautiful dainty pink flowers the bees love. I've grown about a dozen varieties over the years with the creeping thymes such as wooly thyme, elfin thyme, and caraway thyme taking top honors in my garden as the most visual (at one point in time, I grew herbs strictly for aesthetics in the garden and seldom used them in cooking). Today, I have one faithful variety I use for drying and cooking, "french thyme" (Thymus Vulgaris). I previously wrote about drying thyme at this link.
After about three years, thyme plants grow leggy and "woody" at the base. You get long stems with a cluster of leaves at the end. This is the signal for you to work your magic and create new plants. It's very simple and there are two methods. Often times, the end of the branch will start to attach itself at a new location and roots will form. If you see visible roots, you can snip the stem off at the base and plant that new plant elsewhere keeping it moist until it roots and starts growing. If the long stems have not yet grown roots, you can simply bend the stem into the ground next to the plant, cover it with dirt, and new roots will grow starting a new plant. this is a method called layering and is described more fully in the below link. That new plant can be transplanted in a few weeks/months. Mother Earth News did a wonderful article several years back about Thyme. You can read it here. I still enjoy running my fingers through the many scents of thyme at a nursery. It's astounding how caraway thyme really smells like caraway seed, or lemon thyme is so lemony. Enjoy your thyme!