Powershell - Reboot and Continue Script

There is a great article on TechNet from the Hey, Scripting Guy series that goes over a situation very similar to what you are describing: Renaming a computer and resuming the script after reboot. The magic is to use the new workflows that are part of version 3:

workflow Rename-And-Reboot {
  param ([string]$Name)
  Rename-Computer -NewName $Name -Force -Passthru
  Restart-Computer -Wait
  Do-MoreStuff
}

Once the workflow has been declared (you don't assign it to a variable), you can call it as though it were a regular cmdlet. The real magic is the -Wait parameter on the Restart-Computer cmdlet.

Rename-And-Reboot PowerShellWorkflows

Source: https://devblogs.microsoft.com/scripting/powershell-workflows-restarting-the-computer/

If PowerShell v3 or later isn't an available choice, you could break your existing script into multiple smaller scripts and have a master script that runs at startup, checks some saved state somewhere (file, registry, etc.), then starts executing a new script to continue on where appropriate. Something like:

$state = Get-MyCoolPersistedState
switch ($state) {
  "Stage1" { . \Path\To\Stage1.ps1 ; break }
  "Stage2" { . \Path\To\Stage2.ps1 ; break }
  "Stage3" { . \Path\To\Stage3.ps1 ; break }
  default { "Uh, something unexpected happened" }
}

Just be sure to remember to set your state appropriately as you move through your smaller scripts.

The above answer is true, but it will only apply to remote execution of powershell scripts. According to the windows web portal, the way to have your locally running script resume from where it left off after the local machine restarted is like so:

workflow Resume_Workflow
{
    .....
    Rename-Computer -NewName some_name -Force -Passthru
    Restart-Computer -Wait
    # Do some stuff
    .....
}
# Create the scheduled job properties
$options = New-ScheduledJobOption -RunElevated -ContinueIfGoingOnBattery -StartIfOnBattery
$secpasswd = ConvertTo-SecureString "Aa123456!" -AsPlainText -Force
$credential = New-Object System.Management.Automation.PSCredential ("WELCOME\Administrator", $secpasswd)
$AtStartup = New-JobTrigger -AtStartup

# Register the scheduled job
Register-ScheduledJob -Name Resume_Workflow_Job -Trigger $AtStartup -ScriptBlock ({[System.Management.Automation.Remoting.PSSessionConfigurationData]::IsServerManager = $true; Import-Module PSWorkflow; Resume-Job -Name new_resume_workflow_job -Wait}) -ScheduledJobOption $options
# Execute the workflow as a new job
Resume_Workflow -AsJob -JobName new_resume_workflow_job

Note that the [System.Management.Automation.Remoting.PSSessionConfigurationData]::IsServerManager flag should be set to true only if the workflow actions are meant to execute locally after the restart.