First port of call is clearly a good, proven DOF like Campos, Berta, Paratici/Marotta, Zorc (unlikely to leave Dortmund but might for big money), Ragnick/Mitchell or Overmars. Webber of Norwich seems another good option if we want someone UK based.
For me, we need to go all out for the next big thing in management and grow with them. I think many of us fans will be prepared to be patient if we see progress, exciting football and a clear strategy in place. I think Ole's been implementing a culture change and the transfers in were pretty good this Summer, albeit there certainly weren't enough and more deadwood needed shifting before they further depreciated in value. However, on the pitch, there really doesn't seem to be any discernible plan manifesting and I can't really think of any player that's improved besides McTominay. I also don't feel Solskjaer has shown enough prior to taking the wheel. I'd hope he goes before the huge amount of affection we hold for him is tarnished irreparably.
Nagelsmann is the outstanding candidate in my view. He'll be hard to shift out of RB Leipzig; especially so soon after taking the reigns. Graham Potter of Brighton interests me a lot. I like Eddie Howe but Bournemouth are far too frequently trounced for my liking and I worry about his ability to organise a defence. Ten Hag is an obvious candidate. Marco Rose is exciting and Gladbach are top of the Bundesliga so far. However, I'm unsure about his preferred formation (very centrally heavy with a midfield diamond I believe).
A few under the radar options -
Philippe Clement: Started management career very impressively at Waasland-Beveren. Led Genk to 2 consecutive league titles before moving to Club Brugge where he's currently top of the league and forced Real Madrid to a draw. Has played in England for Coventry.
Joao Pedro Sousa: Arguably the brains behind Marco Silva's rise. Now he's gone solo to manage Famalicao, Silva is failing. However, Famalicao are so far currently undefeated and top of the Portuguese league above Porto and Benfica.
Diego Martinez: The 38 year old has surprisingly led Granada into La Liga and now up to 4th
Mark van Bommel: Took PSV to 3 points off an incredible Ajax team last season and now is neck on neck with them points wise with IMHO a far weaker squad
Adi Hutter: Frankfurt manager. Took them to 7th last year and the EL Final. Led Young Boys to their first Swiss league title in 32 years. Won the double at Salzburg. Took Grodig to the Europa League. A very strong managerial record.