The name comes from slavery: "abolitionists" were those who argued that any form of slavery was immoral, and must be eradicated immediately. This is opposed to general "anti-slavery activists" who wanted a gradual phase out of some, or perhaps all, slavery.
An under-appreciated point is that very few anti-slavery activists were actually abolitionists, and the term was more often used to slander the Republican party. Just as Obama increased government funding for health care and was labeled a "socialist," Abraham Lincoln ran on a platform of slightly improving the well-being of slaves and was labeled an "abolitionist." (He supported, for example, the Corwin Amendment which would have amended the Constitution to ensure slavery in states where it already existed - hardly an abolitionist stance.)
Even when Lincoln had the war-time power to issue "Executive Orders" (i.e. laws which didn't have to pass Congress), his famous Emancipation Proclamation did not, despite your 8th grade history textbook, made slavery illegal. It only freed about 75% of current slaves, and said nothing about the practice of slavery itself. (Slavery was later made illegal by the 13th amendment.)
The moral of this history lesson is that change is slow, and often requires making deals with the devil.
But I digress. The point I wanted to make is that there really is no way to work towards improving animal welfare without working towards abolition. This comes from what economists call the Law of Demand:
Consumers buy more of a good when its price decreases and less when its price increases.
This law is self-evident enough that I don't think it requires any argumentation. To make clear its relevance, we can rewrite it:
Consumers buy less meat when its price increases, due to increased regulations in the form of larger cages, more frequent veterinary inspections, etc.
We can in fact directly calculate the "abolition coefficient" of a welfare change. The elasticity of demand is a measure of how demand for a good changes relative to its price. Because we live in a country bloated with agricultural subsidies, armies of economists are employed to predict how demand for various animal products changes with price. Here's a table taken from Huang:
Elasticity is a measurement of percent change. So for beef, a 10% increase in price leads to 6.1% fewer cows on factory farms.
The precise calculation of how much a given law will change the lives of animals is of course complex. But the important thing to note is that these elasticities are always negative - more expensive meat always leads to less meat. Always.
(An important consideration is to wonder if increasing the price of chicken will just drive people to eating beef instead, a rather dubious gain. You can see from e.g. Eales and Unnevehr that, while this does happen to some degree, people do substitute vegetables for meat, meaning that there is a real gain.)