Minimize Spotify to tray on startup by running this script on login with Task Scheduler. Just make sure that Spotify is set to run on startup and close to tray. You may need to increase the sleep time parameter on line 65 if your Spotify window opens too slowly for the script to catch.

spohk.ps1

function Set-WindowState {
	<# .LINK https://gist.github.com/Nora-Ballard/11240204 #>

	[CmdletBinding(DefaultParameterSetName = 'InputObject')]
	param(
		[Parameter(Position = 0, Mandatory = $true, ValueFromPipeline = $true)]
		[Object[]] $InputObject,

		[Parameter(Position = 1)]
		[ValidateSet('FORCEMINIMIZE', 'HIDE', 'MAXIMIZE', 'MINIMIZE', 'RESTORE',
					 'SHOW', 'SHOWDEFAULT', 'SHOWMAXIMIZED', 'SHOWMINIMIZED',
					 'SHOWMINNOACTIVE', 'SHOWNA', 'SHOWNOACTIVATE', 'SHOWNORMAL')]
		[string] $State = 'SHOW'
	)

	Begin {
		$WindowStates = @{
			'FORCEMINIMIZE'		= 11
			'HIDE'				= 0
			'MAXIMIZE'			= 3
			'MINIMIZE'			= 6
			'RESTORE'			= 9
			'SHOW'				= 5
			'SHOWDEFAULT'		= 10
			'SHOWMAXIMIZED'		= 3
			'SHOWMINIMIZED'		= 2
			'SHOWMINNOACTIVE'	= 7
			'SHOWNA'			= 8
			'SHOWNOACTIVATE'	= 4
			'SHOWNORMAL'		= 1
		}

		$Win32ShowWindowAsync = Add-Type -MemberDefinition @'
[DllImport("user32.dll")]
public static extern bool ShowWindowAsync(IntPtr hWnd, int nCmdShow);
'@ -Name "Win32ShowWindowAsync" -Namespace Win32Functions -PassThru

		if (!$global:MainWindowHandles) {
			$global:MainWindowHandles = @{ }
		}
	}

	Process {
		foreach ($process in $InputObject) {
			if ($process.MainWindowHandle -eq 0) {
				if ($global:MainWindowHandles.ContainsKey($process.Id)) {
					$handle = $global:MainWindowHandles[$process.Id]
				} else {
					Write-Error "Main Window handle is '0'"
					continue
				}
			} else {
				$handle = $process.MainWindowHandle
				$global:MainWindowHandles[$process.Id] = $handle
			}

			$Win32ShowWindowAsync::ShowWindowAsync($handle, $WindowStates[$State]) | Out-Null
			Write-Verbose ("Set Window State '{1} on '{0}'" -f $MainWindowHandle, $State)
		}
	}
}

Write-Host "Waiting for Spotify to start..." 
while ((Get-Process | Where-Object { $_.Name -eq "Spotify" }).Count -eq 0) {}
Start-Sleep 1
Get-Process Spotify | Set-WindowState -State Hide