What modifications to Big Bang cosmology have been made to account for all its problems?

What do you mean 'all' of its problems?

So far, most of the problems that Big Bang cosmology has had has been with the quality of the data that has been used to test it, not with anything fundamental in the theory itself.

The only 'problem' that the general public has ever heard about has been the 'age' issue in which globular cluster ages have looked like they were longer than the age of the universe based on the rate of expansion of the universe today. A universe cannot be younger than its contents, and both the media and some astronomers played this up as a 'MAJOR EMBARRASSMENT' to Big Bang cosmology. Then, the Hipparchos satellite returned its results on the distances to over 100,000 stars including 90 that are used for cosmological distance calibration. The verdict was that, from this data, the universe was actually 10 percent older, and the globular cluster stars were 10 percent younger than previously determined.

Then last year, astronomers used distant supernovae to discover than the universe hasn't been steadily expanding and slowing down, but it has been expanding and speeding up! A new ingredient to the intergalactic vacuum, already included in Big Bang cosmology, was confirmed. Now, the age of the universe is comfortably much older than the ages of the oldest stars to the extent that we know how to estimate stellar ages accurately for such a population of stars.

The bottom line is that instead of a cosmological 'train wreck' for Big Bang cosmology, it has once again proved itself bang-on the money. Right now, the WMAP cosmic background data strongly suggests that a Big Bang cosmology with a 'flat' universe fits the data, and the cosmological constant Dark Energy term accounts for about 73 % of the critical density. Dark Matter accounts for 23% and baryonic matter (stars, gas etc) accounts for 4%. The age of the cosmos is now 13.7 billion years with an uncertainty of about 200 million years.

The bottom line is that Big Bang Cosmology still RULES!