For me, the answer is not partial detachment (implying partial attachment), the retention of a certain degree of attachment, but with the attachment brought down to a balanced level.
It really is disinterest.
Disinterest firstly means that my reliance is placed entirely on God (and therefore not at all on the other person(s)). This means that my happiness is based solely on my relationship with God, and I do not need people (a) to be there and (b) to act in a certain way for me to be OK.
Partial attachment is really attachment. Either I’m on the ship in the storm or I’m on shore. If I’m even partly on that ship, I’m really entirely on that ship. Anything else is illusion.
But this disinterest is not indifference.
The corollary of the dependence on God is God entrusting me with the task of loving others, which means not sentimental attachment or interested dependence but a will for others’ good. This means I will act in the benefit of others without attachment to whether and how the benefit arises from the action.
This love of others arises automatically out of the love given to me by God. It is out of that surplus that I extend in love towards others. It is not a bargaining chip in the exchange with the other person, the quid for their quo. I am not giving something up to get something back. Personal(ised) sadness or disappointment about someone else, even if it appears to be on their behalf, at root is the grievance that I will not get out what I put in. If my reward is the giving itself, it matters not what the result of the giving is.
The thing about love is that most seeds scattered do die, are picked off by birds, dry up in the sun, but a few shoot down roots and produce a thousand-fold harvest. The game always pays off at the level of the planting as a whole. Not only is the Master of the farm happy with His labourer, but a harvest is always produced.
Attachment is the labourer picking a particular seed he decides will take and produce a plant and harvest. Unhappiness is confrontation with the disappointment that the seed will never produce a crop. Most seeds, individually, do not. But seeds, in the aggregate, do.
Dialling down attachment is not the answer. God-reliance is.