  The simulated tempering (ST) is an important method to deal with systems
whose phase spaces are hard to sample ergodically. However, it uses accepting
probabilities weights which often demand involving and time consuming
calculations. Here it is shown that such weights are quite accurately obtained
from the largest eigenvalue of the transfer matrix -- a quantity
straightforward to compute from direct Monte Carlo simulations -- thus
simplifying the algorithm implementation. As tests, different systems are
considered, namely, Ising, Blume-Capel, Blume-Emery-Griffiths and Bell-Lavis
liquid water models. In particular, we address first-order phase transition at
low temperatures, a regime notoriously difficulty to simulate because the large
free-energy barriers. The good results found (when compared with other well
established approaches) suggest that the ST can be a valuable tool to address
strong first-order phase transitions, a possibility still not well explored in
the literature.
