Get a CD4000B series inverter, if you can find one that is a single stage
even better. Drive the input via a resistor and feed back the output to
the input via a capacitor. The dV/dt now opposes the input signal to make
a linear ramp.
Not the greatest scheme, since the inverter will spend a lot of time in
cross-conduction, but it's an approach. Be sure you get the good
CD4000 with a high voltage rating, there are weaker subfamilies.
Since you have all those other inverters for free, you might make
a phase shift oscillator and ditch the 555.