Do both sentences mean the same thing?
1) Had he gone to a hospital sooner, more likely he might have survived.
2) Had he gone to a hospital sooner, he would more likely have survived.
Neither one is correct, I would suggest
Had he gone to a hospital sooner he would/might have been more likely to survive
For me a hospital or the hospital are both correct as it could be referring to any hospital or a specific hospital
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The 2nd sentence is more correct but it should be "to the hospital" not to "to a hospital".