I'm a manager - Rusty and EnglishGirl have both made good points. Remember this is not about what you actually think or do, but how it is perceived by other people, including other managers, in your firm.
Managing can be a really fine balance; whilst making changes which are positive for morale can be great, you also need to consider what that looks like to an outsider. Did you get buy-in for your plans from your (temporary) line manager? If not then be careful that this doesn't look like empire building whilst someone is off sick. As PP have said there may be good reasons why they appointed her instead of you.
It's also fab that your team are giving good feedback about you, but again if you aren't engaging with other managers to provide transparency about your plans, and also making it clear how this benefits your line manager who is absent plus the firm, then it is going to look like you are building bridges with your direct reports to alienate them from the other manager and align their loyalty to you. Again I'm not saying that this is your intention, but this is what it may look like.
You need to decouple yourself from the other manager and stop thinking about her as a rival. In your working life this will probably not be the last time that you go up against another colleague for a promotion. Losing out and being disappointed is something that you need to learn how to manage professionally. So switch the focus on to your personal development; if you haven't done so already then ask for feedback about why you were unsuccessful, then take this and form development objectives from it. Continue to ask for feedback about your progress against these objectives - make it clear you are serious about your professional growth. Assuming your firm is reasonably fair about these things then this approach should pay dividends. However it is contingent about how you behave - and are seen to behave - during that period as well.
I saw an example of this at an old firm. Two reasonably well matched people went for an internal promotion and one lost out. The one who was unsuccessful was very disappointed - understandably so. Unfortunately they chose to be quite vocal about this to the rest of the team and other colleagues. They also drew comparisons on a regular basis about how they would have done things differently and it created a chasm in the team with some people siding with him, and others trying to not get involved.
What this individual didn't know, was that they had missed out on the promotion by a couple of points on the benchmarking score. The person who was successful had the edge on some technical experience, but both candidates were strong. So whilst they didn't get the job, they'd been earmarked for a future managerial position which was due to be created in the next 3-6 months as the organisation was expanding. This was strategic planning and therefore confidential information which was restricted only to the people involved in the project, which included the hiring manager. But the behaviour of the individual in the months after the interview made it clear that they were not suitable for a management role - so when the next position was created a few months later, they weren't even offered an interview for it.
Moral of the story is that just because you can't see an open door right now, doesn't mean that it doesn't exist or that it's not going to be created in the future. The way that an unsuccessful candidate behaves in the months after losing out, is often closely watched by senior managers because they want to see how that person will respond and if they will step up to the challenge.