In this work, we study linear error-correcting codes against adversarial insertiondeletion (insdel) errors, a topic that has recently gained a lot of attention. We focus on two different settings -codes over small alphabets and Reed-Solomon codes.Linear codes over small fields: We construct linear codes over F q , for q = poly(1/ε), that can efficiently decode from a δ fraction of insdel errors and have rate (1 − 4δ)/8 − ε. We also show that by allowing codes over F q 2 that are linear over F q , we can improve the rate to (1 − δ)/4 − ε while not sacrificing efficiency. Using this latter result, we construct fully linear codes over F 2 that can efficiently correct up to δ < 1/54 fraction of deletions and have rate R = (1 − 54 • δ)/1216. Cheng, Guruswami, Haeupler, and Li [CGHL21] constructed codes with (extremely small) rates bounded away from zero that can correct up to a δ < 1/400 fraction of insdel errors. They also posed the problem of constructing linear codes that get close to the half-Singleton bound (proved in [CGHL21]) over small fields. Thus, our results significantly improve their construction and get much closer to the bound.Reed-Solomon codes: We prove that over fields of size n O(k) there are [n, k] Reed-Solomon codes that can decode from n − 2k + 1 insdel errors and hence attain the half-Singleton bound. We also give a deterministic construction of such codes over much larger fields (of size n k O(k) ). Nevertheless, for k = O(log n/ log log n) our construction runs in polynomial time. For the special case k = 2, which received a lot of attention in the literature, we construct an [n, 2] Reed-Solomon code over a field of size O(n 4 ) that can decode from n − 3 insdel errors. Earlier construction required an exponential field size. Lastly, we prove that any such construction requires a field of size Ω(n 3 ).