Does the Bondy-Chvátal theorem have algorithmic applications beyond Ore's theorem?
Other way than Bondy-Chvátal theorem is brute force search. However brute force search isn't an efficient way of solving this problem - there can be an exponential number of possibilities to test.

You won't have to test so many possibilities, if you will try following algorithm:

http://arxiv.org/pdf/1405.6347.pdf

The main principle of algorithm is "to think over" which paths(possibilities) should be checked than check many wrong paths.

Algorithm can remove unnecessary edges from graph and test when Hamiltonian cycle can't exist in graph.

Algorithms implementation:

http://findinghamiltoniancycle.codeplex.com/