When it's warm, you are better off just not having a base layer. There's not too many brands that I know of that use good performance fabrics that are full body; most people want separates. Just looking around a bit, I did see some available from a few companies. Just make sure to avoid your old-school cotton union suits, etc.
When looking at fabrics, I'd aim for something relatively thin. You don't want any extra insulation during the warm months, and during the cold months, your insulation should be provided by mid-layers. The main job of a base layer is moisture management. Merino wool is super popular these days as a cold and warm weather fiber for base layers. Despite being wool, it's actually not that warm, but it is really soft, and has good moisture handling. If I had no gear already, and had to buy new stuff, I'd get a merino long underwear and top. Wear both in the winter and just the top in warmer weather. The problem with merino is it's fairly delicate, so you'd want to make sure to get a decent brand, and follow all care instructions well.
Polypropylene has a lot of good characteristics (moisture wicking, durability, cost), but it tends to really hold onto body odor (like even after washing). Polyester has similar characteristics to polypropylene, but without the stink. There's a million other synthetics, but knowing that something is made from polyester, for example, doesn't tell you much cause there's many ways to create synthetic fibers, and many ways to create fabrics from those fibers.
You don't have to spend a fortune to get good stuff, just avoid any "5 letters, all caps" brands on Amazon. Also, avoid any cotton and anything fleece, cause that's trying to be a mid layer, not a base layer