From your description, It sounds like a muscle imbalance issue....

My xplanation will sound very simplistic, however having a rigid thoracic spine might mean instabilities in your lumbar region. Having one side affected more than the other suggests an imbalance issue, bracing using your transversus abdominis stabilizes your lumbar spine making it it less mobile as you move your hip into flexion or abduction limited the excessive movement between the lumbar and thorax...

you might have a stiff hip as well...

You definitely have to continue with your core stability exercises, if you are training with a physio consider using an exercise ball, consider exercises to improve the range in your thoracic spine.

Also consider including some of the core stability work in a more functional exercise such as walking

if neither of these are helpful in the longterm, consider getting a lumbar support and use this when you are going on long distance walks alone...

Cheers