You are currently outside of US. You can return on H-4 visa and file for H-4 to H-1 COS. If you want to change employers, then this needs to be accompanied w/ H-1 transfer (cap-exempt H-1 petition), else it may or may not accompany H-1 extension (depending upon when the current petition expires).
You can start working on H-1 once this COS gets approved. Typical processing time is 2-3 months assuming no RFE is issued.
The other option is to go for H-1 visa stamping and then travel on that visa stamp to work on H-1 from Day 1.