I have some more thoughts about using connections and networking effectively.
It's not fair that being well connected helps you get into competitive career paths, but it is unfortunately the reality.
But connections don't have to be close connections to count.
What I would suggest is this. Talk to your DD about what she might be interested in. If there's something she would like to get some experience in but doesn't know how, think about whether you know anyone in that field or in a related field. If you don't, try one more degree of separation. Talk to people you know well and ask them if they know anyone in the relevant field. Once you've found a connection or potential connection in that field, ask them or your mutual friend whether it would be OK if your daughter contacts them. (She can also do this with her own adult connections if she knows them well enough, for example, if she's been friends with someone for a long time and knows their parents well, she might feel able to ask her friend's parents if they know anyone and if it would be OK to contact them.)
Once she's got the green light to contact someone, it should come directly from her. In my experience people are often willing to help a polite, motivated young person, but will probably find it irritating if it's their mum or dad trying to organise everything. Once the contact has been established, she should have some clear ideas about what she wants to get out of it. As a starting point, her first priority should be to ask some questions about what it is really like to work in that field. Not basic questions, but questions which show that she has already done some research herself and has a good idea about what the job involves but now has more specific questions which she can't find the answer to from Googling, or which relate more to that person's specific personal experience. She can then end the exchange by asking for advice about getting work experience in the field, which may lead to an offer of job shadowing or being put in touch with another contact who may be able to help.
My husband and I are in our late 30s. I've been in the same job for a long time and got my current job the conventional way, through a recruiter, after having got my first job through a graduate recruitment scheme and my second job through an open recruitment process with a large employer. So I have not yet changed jobs through networking. But I have recently received unsolicited approaches from other employers because my name has been given to them by someone I've previously worked with.
My husband, on the other hand, changed jobs a couple of years ago after his previous career came to the end of its natural lifespan. After months without success, we both started talking to anyone we could possibly think of in our entire extended network, and realised the power of networking. A chance conversation with one of the senior executives I work for led to her offering to put my husband (who she had never met) in touch with an acquaintance of hers who had previously been in the same field, and when my husband contacted him he was very happy to meet up and offer advice even though there were three degrees of separation between them.
It sounds silly but I wish I had figured out that this is how the world works when I was 21, not now I am nearing 40. I've done more than alright for myself (with my English degree!) but sometimes I wonder what other opportunities might have presented themselves if I'd been a bit more savvy about this sort of stuff at a young age.
Another silly but important thing. Your network grows and grows with time and it's important to remember who people are. I am terrible with names and faces and I have an awful habit of being introduced to someone and having a conversation with them and then instantly forgetting what their name is and what they do. Now every time I meet a new person in a professional context I spend at least 10 minutes looking them up on LinkedIn or elsewhere, familiarising myself with their name, face and job function, whether they're new to the organisation or a long standing employee I'm just meeting for the first time, what team they're in, where they sit within the organisational hierarchy, and anything else they've told me such as the fact that they have three children or they're an Arsenal fan. It's not stalking, it's essentially trying to cement my interaction with that person in my memory so that rather than forgetting who they are and having an awkward moment the next time we run in to each other, I can smile and ask how they are, confident that I'm not going to get their name wrong and that I might even be able to make small talk with them about something we've previously discussed. Essentially, you want to make your interactions with people meaningful, not only to avoid rudeness or embarrassment, but to make sure that the connections you are making are strong rather than weak. This is also something I wish I had learned at the beginning of my career rather than halfway through it. It's also something that applies equally to people who are or appear to be low down the food chain rather than at the top of it. Certainly in my profession, being rude to admin or support staff is the quickest way for a new grad to get a bad reputation. In the kind of organisation you want to work for, the person in charge knows everyone by name, from their senior management colleagues to the person who makes the tea, and people at the start of their careers would be wise to take note of that and learn from it.