Let's look at the two main approaches--front to back, and bottom to top--in more detail for some hints of which might work best in a given situation.
For me, bottom to top is the best method for the writer who must work alone, soon after the initial writing, with little or no input from alpha readers or editor. It requires reading clear through a minimum of three times: once for the design, once for the construction (how the design was executed) and once for finish, but if done well it produces a final version that is completely coherent (within the writer's own ability to see the work clearly) and feels "organic." Each level of revision is done sequentially, front to back (since everything early in the work affects later sections.)
In the design level reading, you're looking for design flaws in what I call "deep logic" (why things work as they do), motivation, actions, etc. In other words, does the story make sense in terms of your story-universe's reality? Do the characters' actions make sense in terms of the personalities you've given them? Are events in the right order? Do causes come before effects, and are there causes for each effect, and effects from every cause? Even if you don't outline (I don't), by the end of a story or book you should be able to outline it, and the outline should make sense. Does the tone make sense (if you set out to write a funny love story, is it still a funny love story all the way through, or did part of it suddenly turn into a political rant?) Do the emotional highs and lows come where you need/want them, and is the final climax/emotional payoff really the big one? (It's easy to overbuild an earlier one and have insufficient "oomph" in the end.) Since nothing else will be right if the design isn't right, you really do need to work on that first. The "outline" doesn't need to look like the outlines you learned at school, or the one that more organized minds can do up for a book proposal...but it should have "flow" in terms of cause and effect, logic, emotional push/pull, suspense/revelation, etc. If you find a design flaw, note what you wanted to have accomplished there, as well as what you hope will accomplish it. This will help in tearing out and reworking the right part. Design fixes usually require removing and adding material--sometimes a lot of it. You are putting the structure of the story back into the fire, making it malleable again, even completely remelting/remaking it. Write all those new additions in first-draft mode (in other words, you can do it fast and rough if you first-draft that way.)
Once the design level has been reworked, the next step in bottom to top revision is the construction level. Again, this will take at least one reading straight through with construction problems only in mind. This is where you deal with rough transitions, clumsy sentences, out-of-order sentences (which happens to me a lot--when I'm writing fast I may reverse sentences between my brain and my fingers and scenes that don't do what they're meant to do.) You can also catch additional extraneous scenes in this step...something that seemed important in an earlier draft may now not fit, or be too long, because what you added in the design level revision changed things. Some writers can do quite a bit of the finish work at this level (it depends on how good the construction was in the first place) but it will still need at least one detailed check on that afterwards.
The finish passes--however many there are--deal with all the obvious surface details. Spelling, of course. Words repeated too often, too close together. Personal quirks--starting sentences with gerunds, or writing in too obvious a rhythm or the use of a pet word or phrase that shows up in everything you write. "Fossils" from earlier drafts (the sentence into which you put a new detail, making it ungrammatical, for instance.) This is the best time to read the work aloud (even mumbling quietly to yourself) because that will reveal flaws you didn't see as your eye skimmed the familiar material. You will find the best word, not just the adequate word; you will add the jeweler's rouge to the tools you've used so far, and the story will shine, then sparkle.
And because it was worked from the raw material up--because it's all cast as one piece and worked as one piece--it will look and sound and feel all one piece to the reader.
Next time--the front to back approach. But note: this was front to back too, only one layer at a time.