This paper analyzes wireless network control for remote estimation of linear time-invariant dynamical systems under various Hybrid Automatic Repeat Request (HARQ) packet retransmission schemes. In conventional HARQ, packet reliability increases gradually with additional packets; however, each retransmission maximally increases the Age of Information and causes severe degradation in estimation mean squared error (MSE) performance. We optimize standard HARQ schemes by allowing partial retransmissions to increase the packet reliability gradually and limit the AoI growth. In incremental redundancy HARQ, we optimize the retransmission time to enable the early arrival of the next status updates. In Chase combining HARQ, since packet length remains fixed, we allow retransmission and new updates in a single time slot using non-orthogonal signaling. Non-orthogonal retransmissions increase packet reliability without delaying the fresh updates. We formulate bi-objective optimization with the proposed variance of the MSE-based cost function and standard long-term average MSE cost function to guarantee short-term performance stability. Using the Markov decision process formulation, we find the optimal static and dynamic policies under the proposed HARQ schemes to improve MSE performance further. The simulation results show that the proposed HARQ-based policies are more robust and achieve significantly better and more stable MSE performance than standard HARQ-based policies.