In the user-centric cell-free massive MIMO (UC-mMIMO) network scheme, user mobility necessitates updating the set of serving access points to maintain the user-centric clustering. Such updates are typically performed through handoff (HO) operations; however, frequent HOs lead to overheads associated with the allocation and release of resources. This paper presents a deep reinforcement learning (DRL)-based solution to predict and manage these connections for mobile users. Our solution employs the Soft Actor-Critic algorithm, with continuous action space representation, to train a deep neural network to serve as the HO policy. We present a novel proposition for a reward function that integrates a HO penalty in order to balance the attainable rate and the associated overhead related to HOs. We develop two variants of our system; the first one uses mobility direction-assisted (DA) observations that are based on the user movement pattern, while the second one uses history-assisted (HA) observations that are based on the history of the large-scale fading (LSF). Simulation results show that our DRL-based continuous action space approach is more scalable than discrete space counterpart, and that our derived HO policy automatically learns to gather HOs in specific time slots to minimize the overhead of initiating HOs. Our solution can also operate in real time with a response time less than 0.4 ms.