Most women do have a mild anemia in pregnancy. This occurs because our bodies are producing a lot more blood and blood cells to provide the mom and the baby oxygen and other nutrients. What happens is that the amount of fluid is larger than the amount of cells, so pregnant women seem a bit anemic, even when they are taking vitamins and iron. It's almost impossible for the vast majority of people to get too much iron.
Though most prenatal vitamins have iron, they also have so many other nutrients that it's difficult for our bodies to absorb it all. An over-the-counter iron supplement is a good idea. Try to take it with a juice that has lots of vitamin C, which aids in absorption. Avoid taking iron with dairy products, as that decreases absorption. One added iron tablet can be taken once, twice, or three times each day. It takes our bodies about three weeks to produce more mature red blood cells. And the more we have, the more energy we'll have after delivery, when you really need it.
Added iron can cause constipation, so be sure to get lots of water, fruits, and veggies to counteract that particular side effect - which can lead to the dreaded hemorrhoids. Here's a tip: Any fruit with a PIT will make you POOP! (Plums, prunes, apricots, cherries, oh my!)