What you describing is the detection of breakpoints in time series due to changes in deterministic trends. This is referred to as Intervention Detection and was first suggested by I.Chang supervised by G.C.Tiao in 1979 (see https://arxiv.org/pdf/1101.0912.pdf) for more on outlier detection. Early work did not include Trend Detection but more recent software advances does so. The idea is that you don't know a priori how many individual points there are in a new trend or the number of "new trends" . Simple software has approached this with the assumption of the length (duration) of the trends/splines , the # of new trends and naively no ARIMA structure ; all to no avail . Most implementations of Intervention Detection ignore seasonal pulses but do have the facility to detect trend changes which must be done in concert with ARIMA structure. Trying to find trend change points without considering ARIMA structure is like fighting with one hand tied behind your back.
Additionally most implementations of Intervention Detection require pre-specification of the ARIMA structure whereas both components need to be found simultaneously, a neat trick that can be rarely found. Additionally if you have user-suggested predictor variables Intervention Detection is also generally unavailable.
I suggest that you simulate an example or 2 or 3 (not toooo complex) and try different software packages that offer Intervention Detection and see how it goes.
I should add that Intervention Detection was also popularized by William Bell https://www.federalpay.org/employees/bureau-of-the-census/bell-william-r and R. Tsay http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html
You might also search on user:3382 TREND DETECTION as I have commented on trend detection before. Particularly Detecting initial trend or outliers might be illuminating