Op-Amp Sample and Hold Circuit Help
The op-amp circuit following the sample and hold is intended to have a gain of two and I think this over complicates things with the mid-rail generator formed by R1 and R2. Try this for size: -


Get rid of R1, R2 and R4 - just remove them and don't connect anything in their place.
Get rid of D1 and short it out - it's just adding an error that is not needed.


Now, if you have 3V on the input and you sample it, 3V will get charged onto C1 and the output from the op-amp will be a buffered 3V - isn't this what you want?

If you do need a gain of two then restore R4 but take it down to GND/0V - this time, with 3V being sampled it'll be 6V at the output. If your maximum input voltage is 4V you'll get 8.8V on the output - except the LF353 is a poor choice these days - it's not rail-to-rail which means you might see 7.5V maximum on the output - it's going to limit to about 1.5V below the 9V supply voltage.

If, in fact you have a 12V rail, then using it as a gain of two is fine because if your input is restricted to 5V it'll produce a 10V output no problem.