There's no reason you can't use Mary liked the place she had picked. It reads just fine.
In English, it's not necessarily common to add short prepositions after a word, unless you are forming a phrasal verb. Phrasal verbs are words that take on new meanings once a preposition is added. Usually, the two-word entry earns its own mention in the dictionary.
If you look at the word pick in the dictionary, for example, you'll see quite a few definitions listed. Collins lists about a dozen. It also lists five phrasal verbs: pick at, pick off, pick on, pick out, and pick up.
If you go examine each of those, you'll find even more ways to use the word pick. For example, pick up has another dozen or so definitions of its own, such as:
- to resume (let's pick up the game where we left off)
- to catch a disease (she picked up a cold last week)
- to receive a broadcast (we won't be able to pick up that station until we get closer to the city)
Now, back to the place Mary picked, that delightful little open-air cafe.
Check out pick out, Definition #1:
pick out (v.) to select for use or special consideration, illustration, etc, as from a group
Assuming Mary had checked three or four luncheonettes, and picked out one of them, you could say picked out.
However, there is also pick, Definition #1:
pick (v.) to select for use or special consideration, illustration, etc, as from a group
So, in this case, the words pick and pick out can be used interchangeably. However, that's not always the case, and we can't simply arbitrarily tack on words. For example,
Mary liked the place she had picked up.
would be wrong. Even though pick up has quite a few definitions, none of them mean "to select from among a group." Mary might decide to pick up the bill at the cafe (Meaning #20 - how gracious of her!). She might also pick up her fork, when the salads arrive (Meaning #12). If the cafe has a gift shop, she might decide to pick up a souvenir on her way out (Meaning #13). But she won't pick up the restaurant –
not unless it's a model restaurant, and she's about to set it down amidst a model village in her living room.